1 Introduction
The 3D non relativistic Vlasov–Maxwell system is an important model in kinetic theory used to describe the time evolution of the phase space distribution function
$f: \mathbb {R}_{+} \times \mathbb {R}^3 \times \mathbb {R}^3 \rightarrow \mathbb {R}$
of a plasma in interaction with its self-generated electromagnetic field
$(\boldsymbol {A}, \boldsymbol {B}): \mathbb {R}_{+} \times \mathbb {R}^3 \rightarrow \mathbb {R}^3 \times \mathbb {R}^3$
. The Cauchy problem associated with the Vlasov–Maxwell system is given by
$$ \begin{align} \begin{cases} \partial_t f + v \cdot \nabla_x f + \frac{q}{m} \left( \boldsymbol{E} + \frac{v}{c} \times \boldsymbol{B} \right) \cdot \nabla_v f = 0 , \\ \partial_t \boldsymbol{E} - c \nabla \times \boldsymbol{B} = - 4 \pi q \int v f dv , \\ \partial_t \boldsymbol{B} + c \nabla \times \boldsymbol{E} = 0 \end{cases} \end{align} $$
with initial conditions
$(f_0, \boldsymbol {E}_0, \boldsymbol {B}_0)$
satisfying
$$ \begin{align} \nabla \cdot \boldsymbol{E}_0 = 4 \pi q\int f_0\,dv \quad \text{and} \quad \nabla \cdot \boldsymbol{B}_0 = 0. \end{align} $$
Here, q and m are the charge and mass of the charged particles and
$c \geq 1$
is the velocity of light. The local-in-time well-posedness of (1) was proven in [Reference Asano5, Reference Degond22, Reference Wollman56, Reference Wollman57, Reference Wollman58]. The global existence of weak solutions with large data was shown in [Reference Diperna and Lions23]. Furthermore, [Reference Asano and Ukai6, Reference Degond22, Reference Brigouleix and Han-Kwan14] show that for an infinite light velocity, that is,
$c\uparrow \infty $
, the solutions of (1) converge to solutions of the Vlasov–Poisson systems.
In this paper we study a regularized version of (1) and prove its emergence in the semiclassical limit from a regularized (fermionic) Maxwell–Schrödinger system in the Coulomb gauge, as given in (9). The regularization we adopt can be viewed as a finite-size requirement on the particles, aligning with the assumptions typically employed for the underlying many-body model of nonrelativistic quantum electrodynamics, specifically the Pauli–Fierz Hamiltonian (see [Reference Spohn55], Section 1.1 and Remark 2.1 below). Our result provides error estimates in weighted norms, with explicit control of the semiclassical limit in terms of the semiclassical parameter
$\varepsilon $
, which plays the role of the Planck constant
$\hbar $
.
1.1 Heuristic discussion
In order to better understand the connection between the Vlasov-Maxwell and the Maxwell-Schrödinger equations and to see how both systems effectively emerge from nonrelativistic quantum electrodynamics let us consider the evolution of N fermions (with nonrelativistic dispersion and without spin) in interaction with the quantized electromagnetic field in the Coulomb gauge. The functional setting for this system is the Hilbert space
$L^2_{\mathrm {{as}}}(\mathbb {R}^{3N}) \otimes \bigoplus _{n \geq 0} \mathfrak {h}^{\otimes _s^n}$
where the subscripts “as” and “s” indicate antisymmetry and symmetry under exchange of variables, respectively. Here,
$\mathfrak {h} = L^2(\mathbb {R}^3) \otimes \mathbb {C}^2$
consists of photon states
$f(k,\lambda )$
with wave number
$k \in \mathbb {R}^3$
and helicity
$\lambda = 1,2$
. Elements of the Hilbert space evolve in accordance to the Schrödinger equation
$$ \begin{align} i \hbar \partial_t \Psi_{N,t} = H_N^{\mathrm{PF}} \Psi_{N,t} \end{align} $$
with
$$ \begin{align} H_N^{\mathrm{PF}} &= \sum_{j=1}^N \frac{1}{2m} \left( - i \hbar \nabla_j - c^{-1} e \eta * \widehat{\boldsymbol{A}}(x_j) \right)^2 + \frac{e^2}{8\pi} \sum_{\substack{j,k=1\\ k\neq j}}^N \eta * \eta * | \cdot |^{-1} (x_j - x_k) + H_f \end{align} $$
being the spinless Pauli–Fierz Hamiltonian. The dispersion of the photons is
$c \left | k \right |$
,
$H_f = \sum _{\lambda =1,2} \int _{\mathbb {R}^3} \hbar \, c \left | k \right | a_{k,\lambda }^* a_{k,\lambda }\,dk$
denotes the energy of the electromagnetic field and
$$ \begin{align} \widehat{\boldsymbol{A}}(x) &= \sum_{\lambda=1,2} \int_{\mathbb{R}^3} c\,\sqrt{\hbar/(2 c \left| k \right|)}\,\boldsymbol{\epsilon}_{\lambda}(k) (2 \pi)^{-3/2} \left( e^{ikx} a_{k,\lambda} + e^{- ik x} a^*_{k,\lambda} \right)\,dk \end{align} $$
is the quantized transverse vector potential. There are two real polarization vectors
$\big \{ \boldsymbol {\epsilon }_{\lambda }(k) \big \}_{\lambda = 1,2}$
which implement the Coulomb gauge
$\nabla \cdot \widehat {\boldsymbol {A}}=0$
by satisfying
$$ \begin{align} \boldsymbol{\epsilon}_{\lambda}(k) \cdot \boldsymbol{\epsilon}_{\lambda'}(k) = \delta_{\lambda, \lambda'} \quad \text{and} \quad k \cdot \boldsymbol{\epsilon}_{\lambda}(k) = 0. \end{align} $$
The pointwise annihilation and creation operators
$a_{k,\lambda }$
and
$a^*_{k,\lambda }$
satisfy the canonical commutation relations
$$ \begin{align} \left[ a_{k,\lambda} , a^*_{k',\lambda'} \right] = \delta_{\lambda, \lambda'} \delta(k - k') , \quad \left[ a_{k,\lambda} , a_{k',\lambda'} \right] = \left[ a^*_{k,\lambda} , a^*_{k',\lambda'} \right] = 0, \end{align} $$
where
$[A,B]:=AB-BA$
is the standard commutator of the operators A and B,
$\delta _{\lambda , \lambda '}$
is the Kronecker delta, and
$\delta (k - k')$
is the Dirac delta.
The real function
$\eta $
describes the density of the fermions, which are consequently not regarded as point particles but as finite-size particles with charge distribution
$e \eta (x)$
and mass distribution
$m \eta (x)$
. The Pauli–Fierz Hamiltonian is obtained by canonical quantization of the Abraham model and known as a mathematical model of nonrelativistic quantum electrodynamics. If the Fourier transform of the charge distributionFootnote
1
satisfies
$\big ( \left | \cdot \right |{}^{-1} + \left | \cdot \right |{}^{1/2} \big ) \mathcal {F}[\eta ] \in L^2(\mathbb {R}^3)$
(an assumption we will also make later for the effective models, see Remark 2.1) it holds that the Hamiltonian is self-adjoint on the domain
$\mathcal {D} \big ( H_{N}^{\mathrm {{PF}}} \big ) = \mathcal {D} \big ( - \sum _{j=1}^N \Delta _{x_j} + H_f \big )$
[Reference Hiroshima28, Reference Matte38, Reference Spohn55]. For a detailed introduction to the model we refer to [Reference Spohn55]. Let us consider equation (3) in the semiclassical mean-field regime by setting
$e = N^{-1/2}$
and
$\hbar = N^{-1/3}$
, and choose
$c=1$
and
$m= 1/2$
for notational convenience. We are interested in many-body wave functions of the form
$\Psi _N = \bigwedge _{j=1}^N \varphi _j \otimes W(N^{2/3} \alpha ) \Omega $
because they describe fermion-photon configurations with little correlations. Here,
$\bigwedge _{j=1}^N \varphi _j$
is a Slater determinant of N orthonormal one-particle wave functions
$( \varphi _j )_{j=1}^N$
,
$\alpha \in \mathfrak {h}$
, and
$W(N^{2/3} \alpha ) \Omega $
is a coherent state of photons with mean photon number
$N^{4/3} \left \| \alpha \right \| _{\mathfrak {h}}^2$
. It is defined in terms of the Weyl operator
$W(N^{2/3} \alpha ) = \exp \Big ( N^{2/3} \sum _{\lambda =1,2} \int _{\mathbb {R}^3} \alpha (k,\lambda ) a_{k,\lambda }^* - \overline {\alpha (k,\lambda )} a_{k,\lambda }dk \Big )$
and the vacuum
$\Omega $
of the bosonic Fock space over
$\mathfrak {h}$
. One expects that the structure of such states is approximately preserved during the evolution (3), and that, in the limit
$N \rightarrow \infty $
, the time-evolved state at time t is approximated by
$$ \begin{align} \Psi_{N,t} \approx \bigwedge_{j=1}^N \varphi_{j,t} \otimes W(N^{2/3} \alpha_t) \Omega. \end{align} $$
Note that this can be understood as a propagation of chaos assumption and that Slater determinants are completely characterized by their one-particle reduced density matrix
$\omega _{N,t} = \sum _{j=1}^N | \varphi _{j,t} \rangle \langle \varphi _{j,t} |$
. If one uses that the action of a quantum field on a coherent state is approximately given by a classical field and that the interaction between fermions in a Slater determinant can be approximated by its mean-field potential plus an exchange term one sees that the one-particle reduced density matrix and the mode function of the electromagnetic field in (8) should be a solution of the regularized (fermionic) Maxwell–Schrödinger system in the Coulomb gauge
$$ \begin{align} \begin{cases} i \varepsilon \partial_t \omega_{N,t}\kern-7pt &= \left[ \left( - i \varepsilon \nabla - \eta * \boldsymbol{A}_{\alpha_t} \right)^2 + K * \rho_{\omega_{N,t}} - X_{\omega_{N,t}} , \omega_{N,t} \right] , \\ i \partial_t \alpha_t(k,\lambda) \kern-7pt&= \left| k \right| \alpha_t(k,\lambda) - \sqrt{\frac{4 \pi^3}{\left| k \right|}} \mathcal{F}[\eta](k) \boldsymbol{\epsilon}_{\lambda}(k) \cdot \mathcal{F}[\boldsymbol{J}_{\omega_{N,t}, \alpha_t}](k) \end{cases} \end{align} $$
with
$$ \begin{align} K &= \frac{1}{4\pi}\, \eta * \eta * | \cdot |^{-1} , \end{align} $$
$$ \begin{align} \boldsymbol{A}_{\alpha}(x) &= (2 \pi )^{- 3/2} \sum_{\lambda = 1,2} \int \frac{1}{\sqrt{2 \left| k \right|}} \boldsymbol{\epsilon}_{\lambda}(k) \left( e^{i k x} \alpha(k,\lambda) + e^{- i k x} \overline{\alpha(k, \lambda)} \right)dk, \end{align} $$
$$ \begin{align} \rho_{\omega}(x) &= N^{-1} \omega(x;x) , \end{align} $$
$$ \begin{align} \boldsymbol{J}_{\omega, \alpha}(x) &= - N^{-1} \left\{ i \varepsilon \nabla, \omega \right\}(x;x) - 2 \rho_{\omega}(x) \eta * \boldsymbol{A}_{\alpha}(x) , \end{align} $$
$$ \begin{align} X_{\omega}(x;y) &= N^{-1} K(x-y) \omega(x;y) , \end{align} $$
semiclassical parameter
$\varepsilon = N^{-1/3}$
and initial condition
$(\omega _{N,t}, \alpha _t) \big |_{t=0} = (\sum _{j=1}^N | \varphi _{j} \rangle \langle \varphi _{j} |, \alpha )$
. In (10d)
$\{A,B\}:=AB+BA$
stands for the anticommutator of the operators A and B, and
$\{A,B\}(x;y)$
denotes the kernel of the anticommutator. Here, K is the regularized potential,
$\boldsymbol {A}_{\alpha }$
is the classical vector potential, and
$\rho _{\omega }$
and
$\boldsymbol {J}_{\omega , \alpha }$
are the density and current density of the fermions. Moreover,
$K * \rho _{\omega }$
and
$X_{\omega }$
represent the mean-field potential and the exchange term, respectively, arising from the direct interaction between the fermions.
While we motivated the emergence of (9) from the Pauli–Fierz Hamiltonian in the case of N fermions at zero temperature, we do not restrict the initial condition of
$\omega _{N,t}$
to rank-N projections, but instead consider a broader class of initial data corresponding to mixed states of fermionic systems at positive temperature.
In the mathematical literature the Maxwell–Schrödinger system appears for
$\varepsilon =1$
, one single particle (
$N=1$
) and without regularization, that is,
$\eta (x) = \delta (x)$
. The second equation of (9) can be written as Maxwell’s equations in the Coulomb gauge
$$ \begin{align} \nabla \cdot \boldsymbol{A}_{\alpha_t} = 0,\quad\quad \left(\partial_t^2 - \Delta\right)\boldsymbol{A}_{\alpha_t} = - \left( 1 - \nabla \text{div} \Delta^{-1} \right) \eta * \boldsymbol{J}_{\omega_t, \alpha_t}. \end{align} $$
In the case of a single particle, the first equation of (9) can be expressed as a Schrödinger equation for the one-particle wave function. For the global well-posedness of (9) with
$N = 1$
,
$\varepsilon = 1$
,
$\eta (x) = \delta (x)$
we refer to [Reference Bejenaru and Tataru9, Reference Nakamura and Wada43] and references therein. Throughout this article we will rely on Proposition 2.1 for the regularized system under consideration.
Note that the rigorous derivation of (9) from the Pauli–Fierz Hamiltonian in the semiclassical mean-field limit of many fermions, previously discussed heuristically, has recently been established in [Reference Leopold33]. With this regard we would like to mention that a rigorous derivation of the Maxwell–Schrödinger system from the Pauli–Fierz Hamiltonian in a mean-field limit of many charged bosons was obtained in [Reference Leopold and Pickl35] (see also [Reference Falconi and Leopold24]) and that the aforementioned fermionic mean-field limit was studied in case of the regularized Nelson model [Reference Leopold and Petrat34], leading to an effective equation in which the fermions linearly couple to a classical Klein–Gordon field. Moreover, let us point out the works [Reference Knowles29, Reference Ammari, Falconi and Hiroshima1] in which the regularized Newton–Maxwell equations are derived from the Pauli–Fierz Hamiltonian in a classical limit. In [Reference Breteaux, Faupin and Payet13] and [Reference Correggi, Falconi and Olivieri21] the existence of a ground state of the Maxwell–Schrödinger energy functional is proven and it’s derivation from the Pauli–Fierz Hamiltonian in the quasi-classical regime is obtained. The scenario where magnetic forces are disregarded, leading to the absence of Maxwell equations in the system, was addressed in [Reference Bach, Breteaux, Petrat, Pickl and Tzaneteas8, Reference Benedikter, Jakšić, Porta, Saffirio and Schlein10, Reference Benedikter, Porta and Schlein12, Reference Chong, Lafleche and Saffirio18, Reference Petrat and Pickl45, Reference Porta, Rademacher, Saffirio and Schlein47], where the Hartree-Fock equation was rigorously obtained in the mean-field regime from the many-body Schrödinger equation. For more details on the above mentioned references we refer to the review [Reference Saffirio52].
Note that (9) depends on N because of the semiclassical parameter
$\varepsilon = N^{-1/3}$
. In order to see the emergence of the Vlasov–Maxwell equations we introduce the Wigner transform of the trace class operator
$\omega _{N,t}$
defined as
$$ \begin{align} W_{N,t}(x,v) = \left( \frac{\varepsilon}{2 \pi} \right)^3 \int_{\mathbb{R}^3} \omega_{N,t} \left( x + \frac{\varepsilon y}{2}; x - \frac{\varepsilon y}{2} \right) e^{- i v \cdot y}dy \end{align} $$
and its inverse, called the Weyl quantization,
$$ \begin{align} \omega_{N,t}(x;y) = N \int_{\mathbb{R}^3} W_{N,t} \left( \frac{x+y}{2} , v \right) e^{i v \cdot \frac{x-y}{\varepsilon}}dv. \end{align} $$
Letting
$\omega _{N,t}$
be a solution of (9), taking its Wigner transform, and performing a formal expansion in powers of
$\varepsilon $
yields, to first order, the following classical magnetic Vlasov equation
$$ \begin{align} \partial_t W_{N,t}(x,v) &= \bigg[ \Big( \nabla K * \rho_{\omega_{N,t}}(x) + 2 \sum_{j=1}^3 (\nabla \eta * \boldsymbol{A}_{\alpha_t}^{j})(x) \big( \eta * \boldsymbol{A}_{\alpha_t}^{j}(x) - v_j \big) \Big) \cdot \nabla_v \nonumber \\ &\qquad + 2 \Big( \eta * \boldsymbol{A}_{\alpha_t}(x) - v \Big) \cdot \nabla_x \bigg] W_{N,t}(x,v) + \mathcal{O}(\varepsilon). \end{align} $$
Together with
$N^{-1} \omega _{N,t}(x;x) = \int _{\mathbb {R}^3} W_{N,t}(x,v)\,dv$
and
$\left \{ i \varepsilon \nabla , \omega _{N,t} \right \}(x;x) = - 2 N \int _{\mathbb {R}^3} \, v W_{N,t}(x,v)\,dv$
, this suggests that in the limit
$N \rightarrow \infty $
, that is,
$\varepsilon \rightarrow 0$
, the Wigner transform of
$\omega _{N,t}$
can be approximated by a phase space function
$f_t$
which satisfies the following transport equation
$$ \begin{align} \begin{cases} \partial_t f_t \kern-7pt&= - 2 \left( v - \eta * \boldsymbol{A}_{\alpha_t} \right) \cdot \nabla_x f_t + \boldsymbol{F}_{f_t, \alpha_t} \cdot \nabla_v f_t , \\ i \partial_t \alpha_t(k,\lambda) \kern-7pt&= \left| k \right| \alpha_t(k,\lambda) - \sqrt{\frac{4 \pi^3}{\left| k \right|}} \mathcal{F}[\eta](k) \boldsymbol{\epsilon}_{\lambda}(k) \cdot \mathcal{F}[\widetilde{\boldsymbol{J}}_{f_t,\alpha_t}](k) \end{cases} \end{align} $$
with
$\boldsymbol {A}_{\alpha _t}$
being defined as in (10b) and
$$ \begin{align} \widetilde{\rho}_{f}(x) &= \int f(x,v) \, dv , \end{align} $$
$$ \begin{align} \boldsymbol{F}_{f, \alpha}(x,v) &= \nabla_x \Big[ K * \widetilde{\rho}_{f}(x) + \left( \eta * \boldsymbol{A}_{\alpha}(x) \right)^2 - 2 v \cdot \eta * \boldsymbol{A}_{\alpha}(x) \Big] , \end{align} $$
$$ \begin{align} \widetilde{\boldsymbol{J}}_{f,\alpha}(x) &= 2 \int \left( v - \eta * \boldsymbol{A}_{\alpha} \right) f(x,v) \, dv \end{align} $$
and initial datum
$(f_0, \alpha _0) \in L^1 \left ( \mathbb {R}^3 \times \mathbb {R}^3 \right ) \times \left ( L^2(\mathbb {R}^3) \otimes \mathbb {C}^2 \right )$
. Note that (15) with
$\eta (x) = - \delta (x)$
is formally equivalent to (1) in the Coulomb gauge with
$c= 1$
,
$q= -1$
and
$m=1/2$
, as shown in Appendix A. For this reason hereafter we refer to (15) as the (regularized) Vlasov–Maxwell system.
1.2 Notation
For
$\sigma \in \mathbb {N}_0$
,
$k \geq 0$
and
$1 \leq p \leq \infty $
let
$W_{k}^{\sigma ,p} \left ( \mathbb {R}^{d} \right )$
be the Sobolev space equipped with the norm
$$ \begin{align} \left\| f \right\| _{W_{k}^{\sigma, p} \left( \mathbb{R}^{d} \right)} = \begin{cases} \left( \sum_{\left| \alpha \right| \leq \sigma} \left\| \left< \cdot \right>^{k} D^{\alpha} f \right\| _{L^p\left( \mathbb{R}^{d} \right)}^p \right)^{\frac{1}{p}} \quad &1 \leq p < \infty , \\ \max_{\left| \alpha \right| \leq \sigma} \left\| \left< \cdot \right>^{k} D^{\alpha} f \right\| _{L^{\infty}\left( \mathbb{R}^{d} \right)} \quad &p = \infty, \end{cases} \end{align} $$
where
$\left < x \right>^2 = 1 + \left | x \right |{}^2$
. In the cases
$\sigma =0$
or
$p=2$
we use the shorthand notation 
and 
. Vectors in
$\mathbb {R}^{6}$
are written as
$z = (x,v)$
so that
$\left < z \right>^2 = 1 + \left | x \right |{}^2 + \left | v \right |{}^2$
and
$D_z^{\alpha } = \left ( \partial / \partial x_1 \right )^{\beta _1} \left ( \partial / \partial v_1 \right )^{\gamma _1} \cdots \left ( \partial / \partial x_3 \right )^{\beta _3} \left ( \partial / \partial v_3 \right )^{\gamma _3}$
with 
and 
such that
$\left | \alpha \right | = \sum _{i=1}^3 \beta _i + \gamma _i$
. For two Banach spaces A and B we denote by
$A \cap B$
the Banach space of vectors
$f \in A \cap B$
with norm
$\left \| f \right \| _{A \cap B} = \left \| f \right \| _A + \left \| f \right \| _B$
. For
$r \in \mathbb {R}$
we define the weighted
$L^2$
-space
$\dot {\mathfrak {h}}_{r} = L^2(\mathbb {R}^3, \left | k \right |{}^{2r} dk) \otimes \mathbb {C}^2$
with norm
$$ \begin{align} \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{r}} &= \left( \sum_{\lambda=1,2} \int_{\mathbb{R}^3} \left| k \right|{}^{2r} \left| \alpha(k,\lambda) \right|^2dk \right)^{1/2} = \left\| |\cdot |^{r} \alpha \right\| _{\mathfrak{h}}. \end{align} $$
Note that the notation
$\mathfrak {h}$
will be used to denote the Hilbert space
$\mathfrak {h}_0$
with scalar product
$$ \begin{align} \left\langle \alpha , \beta \right\rangle = \sum_{\lambda=1,2} \int_{\mathbb{R}^3} \overline{\alpha(k,\lambda)} \beta(k,\lambda) dk. \end{align} $$
It is, moreover, convenient to define for
$r \geq 0$
the Banach spaces
$$ \begin{align} \mathfrak{h}_r &= \left\{ \alpha \in \mathfrak{h} : (1 + \left| k \right|^2)^{r/2} \alpha \in \mathfrak{h} \right\} \end{align} $$
with
$$ \begin{align} \left\| \alpha \right\| _{\mathfrak{h}_r} &= \left( \sum_{\lambda=1,2} \int_{\mathbb{R}^3} \left( 1 + \left| k \right|^2 \right)^r \left| \alpha(k,\lambda) \right|^2dk \right)^{1/2}. \end{align} $$
For a reflexive Banach space
$(X, \left \| \cdot \right \| _{X})$
and
$p \in [1,\infty ]$
we denote by
$L^p(0,T; X)$
the space of (equivalence classes of) strongly Lebesgue-measurable functions
$\alpha : [0,T] \rightarrow X$
with the property that
$$ \begin{align} \left\| \alpha \right\| _{L_T^p X} &= \begin{cases} \left( \int_0^T \left\| \alpha_t \right\| _X^p dt \right)^{1/p} \quad \text{if} \; 1 \leq p < \infty \\ \operatorname*{\mbox{ess sup}}_{t \in [0,T]} \left\| \alpha_t \right\| _{X} \quad \text{if} \; p= \infty \end{cases} \end{align} $$
is finite. Note that
$(L^p(0,T; X), \left \| \cdot \right \| _{L_T^p X})$
is a Banach space. The Sobolev space
$W^{1,p}(0,T; X)$
consists of all functions
$\alpha \in L^p(0,T; X)$
such that
$\partial _t \alpha _t$
exists in weak sense and belongs to
$L^p(0,T; X)$
. Furthermore,
$$ \begin{align} \left\| \alpha \right\| _{W^{1,p}(0,T;X)} &= \begin{cases} \left( \int_0^T \left\| \alpha_t \right\| _{X}^p + \left\| \partial_t \alpha_t \right\| _{X}^p dt \right)^{1/p} \quad \text{if} \; 1 \leq p < \infty \\ \operatorname*{\mbox{ess sup}}_{t \in [0,T]} \left( \left\| \alpha_s \right\| _{X} + \left\| \partial_t \alpha_t \right\| _{X} \right) \quad \text{if} \; p= \infty. \end{cases} \end{align} $$
Let
$\mathfrak {S}^{\infty } \left ( L^2(\mathbb {R}^3) \right )$
be the set of all bounded operators on
$L^2 (\mathbb {R}^3)$
and
$\mathfrak {S}^1 \left ( L^2(\mathbb {R}^3) \right )$
be the set of trace class operators on
$L^2(\mathbb {R}^3)$
. More generally, for
$p\in [1,\infty )$
, we denote by
$\mathfrak {S}^p(\mathbb {R}^3)$
the p-Schatten space equipped with the norm
$\left \| A \right \| _{\mathfrak {S}^p}=(\text {Tr} \left | A \right |{}^p)^{\frac {1}{p}}$
, where A is an operator,
$A^*$
its adjoint and
$\left | A \right |=\sqrt {A^*A}$
. For
$r \geq 0$
let
$$ \begin{align} \mathfrak{S}^{r,1}(L^2(\mathbb{R}^3)) &= \left\{ \omega: \omega \in \mathfrak{S}^{\infty}(L^2(\mathbb{R}^3)), \omega^* = \omega \; \text{and} \; \left( 1 - \Delta \right)^{r/2} \omega\left( 1 - \Delta \right)^{r/2} \in \mathfrak{S}^{1}(L^2(\mathbb{R}^3)) \right\} \end{align} $$
with
$$ \begin{align} \left\| \omega \right\| _{\mathfrak{S}^{r,1}} &= \left\| \left( 1 - \Delta \right)^{r/2} \omega \left( 1 - \Delta \right)^{r/2} \right\| _{\mathfrak{S}^1}. \end{align} $$
The positive cone of the latter one is defined as
$$ \begin{align} \mathfrak{S}_{+}^{r,1}(L^2(\mathbb{R}^3)) &= \left\{ \omega \in \mathfrak{S}^{r,1}(L^2(\mathbb{R}^3)) : \omega \geq 0 \right\}. \end{align} $$
The position operator, that is, the operator that multiplies by x, is denoted by
$\hat {x}$
. We use the symbol C to represent a general positive constant that could vary from one line to another, and it may depend on the parameters k and
$\sigma $
in (17), as well as on the cutoff function outlined in (27) below. Positive constants depending on
$\varepsilon $
are denoted by
$C_{\varepsilon }$
. Additionally, we adopt the notation
$C(f,g)$
for positive constants depending on the enclosed quantities (specifically f and g in this context).
1.3 Organization of the paper
The paper is organized as follows. In Section 2 we present our main result concerning the semiclassical approximation of the Maxwell–Schrödinger dynamics by a solution to the Vlasov–Maxwell equation (see Theorem 2.1). This result builds upon the well-posedness and regularity theory for both Maxwell–Schrödinger equations and for the Vlasov–Maxwell system, which are given in Proposition 2.1 and Proposition 2.2, respectively. We also discuss the physical relevance of the assumptions made on the cut-off parameter (see Remark 2.1 and Remark 2.2) and compare our results with the existing literature. Section 3 collects preliminary estimates that will be used throughout the text. Sections 4, 5 and 6 are devoted to the proofs of Theorem 2.1, Proposition 2.1 and Proposition 2.2, respectively. The paper includes two appendices: in Appendix A, we establish the equivalence between two distinct formulations of the Vlasov–Maxwell system, while Appendix B shows that certain Schatten norms of commutators between the Weyl quantization of a particle distribution and position or momentum operators can be related to weighted Sobolev norms of the particle distribution.
2 Main results
Throughout the paper we assume the cut-off parameter to satisfy the following assumptions.
Assumption 2.1. Let
$\eta : \mathbb {R}^3 \rightarrow \mathbb {R}$
be a real and even charge distribution such that
$$ \begin{align} \eta \in L^1(\mathbb{R}^3) \cap H^{1/2} (\mathbb{R}^3). \end{align} $$
Remark 2.1. We will not study the explicit dependence of the estimates on the charge distribution and for this reason estimate the respective norms (27) by a generic not specified constant. The first condition in (27) requires that the negative and positive parts of the charge distribution are summable. If the charge distribution is purely negative or positive it only assumes that the total charge is finite. The second condition requires a sufficient decay of the Fourier modes with high momenta. Note that our assumptions imply
$\mathcal {F}[\eta ] \in L^{\infty }(\mathbb {R}^3)$
and
$$ \begin{align} \left( \left| \cdot \right|{}^{-1} + \left| \cdot \right|{}^{1/2} \right) \mathcal{F}[\eta] \in L^2(\mathbb{R}^3) , \end{align} $$
where the latter one is usually required to prove the self-adjointness of the Pauli–Fierz Hamiltonian (see Subsection 1.1). Since we require
$\eta $
to be even and summable, our assumptions are slightly more restrictive than needed for the self-adjointness of the Pauli–Fierz Hamiltonian. Note that
$$ \begin{align} \mathcal{F}[\eta](k) = \mathcal{F}[\eta](-k) \quad \text{and} \quad \mathcal{F}[\eta](k) \in \mathbb{R} \quad \forall k \in \mathbb{R}^3 \end{align} $$
because
$\eta $
is real and even.
Remark 2.2. A typical example of a cut-off parameter which satisfies our assumptions is
$$ \begin{align} \frac{e}{\sigma^3 (2 \pi)^{3/2}} e^{- \frac{x^2}{2 \sigma^2}} \quad \text{with} \quad \sigma> 0 , \end{align} $$
which describes extended particles with total charge
$e \in \mathbb {R}$
distributed in a Gaussian fashion.
In this paper we rely on the following well-posedness result for the Maxwell–Schrödinger system, which is proved in Section 5.
Proposition 2.1. Let
$\eta $
satisfy Assumption 2.1. For all
$(\omega _0, \alpha _0) \in \mathfrak {S}_{+}^{2,1} (L^2(\mathbb {R}^3)) \times \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2} $
the Cauchy problem for the Maxwell–Schrödinger system (9) associated with
$(\omega _0, \alpha _0)$
has a unique
$ C \big ( \mathbb {R}_{+}; \mathfrak {S}_{+}^{2,1} ( L^2(\mathbb {R}^3) ) \big ) \cap C^1 \big ( \mathbb {R}_{+}; \mathfrak {S}^{1} ( L^2(\mathbb {R}^3) ) \big ) \times C \big ( \mathbb {R}_{+}; \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2} \big ) \cap C^1 \big ( \mathbb {R}_{+}; \dot {\mathfrak {h}}_{-1/2} \big ) $
solution. The mass and energy
$$ \begin{align} \mathcal{E}^{\mathrm{{MS}}}[\omega, \alpha] &= \text{Tr} \left( \omega \left( - i \varepsilon \nabla - \eta * \boldsymbol{A}_{\alpha} \right)^2 \right) + \frac{1}{2} \text{Tr} \left( \left( \, K * \rho_{\omega} - X_{\omega} \right) \omega \right) + N \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 \end{align} $$
of the system are conserved, that is,
$$ \begin{align} \text{Tr} \left( \omega_t \right) = \text{Tr} \left( \omega_0 \right) \quad \text{and} \quad \mathcal{E}^{\mathrm{{MS}}}[\omega_t, \alpha_t] = \mathcal{E}^{\mathrm{{MS}}}[\omega_0, \alpha_0] \quad \text{for all} \; t \in \mathbb{R}_{+}. \end{align} $$
We will also make use of the following result concerning the solution and regularity theory for the Vlasov–Maxwell system. Its proof is provided in Section 6.
Proposition 2.2. Let
$R>0$
and
$a, b \in \mathbb {N}$
satisfying
$a \geq 5$
and
$b \geq 3$
. For all
$(f_0, \alpha _0) \in H_a^{b}(\mathbb {R}^6) \times \mathfrak {h}_b \cap \dot {\mathfrak {h}}_{-1/2}$
such that
$\operatorname {\mathrm {supp}} f_0 \subset A_R$
with
$A_R = \{ (x,v) \in \mathbb {R}^6 , \left | v \right | \leq R \}$
, system (15) has a unique solution in
$L^{\infty } \big ( \mathbb {R}_{+}; H_{a}^{b}(\mathbb {R}^6) \big ) \cap C \big ( \mathbb {R}_{+}; H_{a}^{b-1}(\mathbb {R}^6) \big ) \cap C^1 (\mathbb {R}_{+}; H_{a}^{b-2}(\mathbb {R}^6)) \times C \big ( \mathbb {R}_{+}; \mathfrak {h}_{b} \cap \dot {\mathfrak {h}}_{-1/2} \big ) \cap C^1 \big ( \mathbb {R}_{+}; \mathfrak {h}_{b-1} \cap \dot {\mathfrak {h}}_{-1/2} \big )$
. The
$L^p$
–norms of the particle distribution (with
$p \geq 1$
) and the energy
$$ \begin{align} \mathcal{E}^{\mathrm{{VM}}}[f, \alpha] &= \int_{\mathbb{R}^6} dx \, dv \, f(x,v) \left( v - \eta * \boldsymbol{A}_{\alpha}(x) \right)^2 + \frac{1}{2} \int_{\mathbb{R}^6} dx \, dv \, f(x,v) K * \widetilde{\rho}_{f}(x) + \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 \end{align} $$
are conserved, that is,
$\left \| f_t \right \| _{L^p(\mathbb {R}^6)} = \left \| f_0 \right \| _{L^p(\mathbb {R}^6)}$
and
$\mathcal {E}^{\mathrm {{VM}}}[f_t, \alpha _t] = \mathcal {E}^{\mathrm {{VM}}}[f_0, \alpha _0]$
for all
$t \in \mathbb {R}_{+}$
.
Our main result is the following
Theorem 2.1. Let
$\eta $
satisfy Assumption 2.1,
$\varepsilon = N^{-1/3}$
,
$\alpha _0 \in \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2}$
and
$\omega _{N,0} \in \mathfrak {S}^{2,1} \big ( L^2(\mathbb {R}^3) \big )$
be a sequence of reduced density matrices on
$L^2(\mathbb {R}^3)$
satisfying
$\text {Tr} \left ( \omega _{N,0} \right ) = N$
and
$0 \leq \omega _{N,0} \leq 1$
. Let
$(\omega _{N,t}, \alpha _t)$
be the unique solution of (9) with initial datum
$(\omega _{N,0},\alpha _0)$
. Let
$\big ( \widetilde {W}_{N,t}, \widetilde {\alpha }_t \big )$
be a solution to (15) with initial datum
$\big ( \widetilde {W}_{N,0}, \widetilde {\alpha }_0 \big )$
, such that
$\widetilde {W}_{N,0} \geq 0$
,
$\widetilde {W}_{N,0} \in W_2^{0,1}(\mathbb {R}^6)$
and verifying
$$ \begin{align} \big( \widetilde{W}_{N}, \widetilde{\alpha} \, \big) \in L_{\mathrm{{loc}}}^{\infty} \big( \mathbb{R}_{+}, H_7^8(\mathbb{R}^6) \big) \times L_{\mathrm{{loc}}}^{\infty} \big( \mathbb{R}_{+}, \mathfrak{h}_5 \cap \dot{\mathfrak{h}}_{- 1/2} \big). \end{align} $$
Moreover, let
$\widetilde {\omega }_{N,t}$
be the Weyl quantization of
$\widetilde {W}_{N,t}$
and
$$ \begin{align} \Xi (t) &= N^{-1} \left\| \sqrt{1 - \varepsilon^2 \Delta} \left( \omega_{N,t} - \widetilde{\omega}_{N,t} \right) \sqrt{1 - \varepsilon^2 \Delta} \right\| _{\mathfrak{S}^{1}} + \left\| \alpha_t - \widetilde{\alpha}_t \right\| _{\dot{\mathfrak{h}}_{-1/2} \, \cap \, \dot{\mathfrak{h}}_{1/2}}. \end{align} $$
Then,
$$ \begin{align} \Xi (t) &\leq \left( \Xi(0) + \varepsilon \widetilde{C}(t) \right) e^{C(t)}. \end{align} $$
Here,
$$ \begin{align} \widetilde{C}(t) &\leq \int_0^t ds \, \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{7}^{j+2}} \left( 1 + \sum_{k=0}^3 \varepsilon^{2 + k} \left\| \widetilde{W}_{N,s} \right\| _{H_{4}^{k+1}} + \sum_{k=0}^4 \varepsilon^k \left\| \widetilde{\alpha}_s \right\| _{\mathfrak{h}_{k+1}}^2 \right) \,, \end{align} $$
$$ \begin{align} C(t) &= C \left( 1 + N^{-1} \mathcal{E}^{\mathrm{{MS}}}[\omega_{N,0}, \alpha_0] + \mathcal{E}^{\mathrm{{VM}}}[\widetilde{W}_{N,0}, \widetilde{\alpha}_0] + \left\| \widetilde{W}_{N,0} \right\| _{L^1(\mathbb{R}^6)} \right)^2 \nonumber \\ &\quad \times \left( \left< t \right> + \int_0^t ds \, \sum_{j=0}^{7} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{7}^{j+1}} \right) \end{align} $$
and C is a numerical constant which depends on the specific choice of the cutoff function
$\eta $
.
Remark 2.3. Notice that (34) is satisfied if
$\big ( \widetilde {W}_{N,0}, \widetilde {\alpha }_0 \big ) \in H_7^{8}(\mathbb {R}^6) \times \mathfrak {h}_8 \cap \dot {\mathfrak {h}}_{-1/2}$
such that
$\widetilde {W}_{N,0} \geq 0$
,
$\widetilde {W}_{N,0} \in W_2^{0,1}(\mathbb {R}^6)$
and
$\operatorname {\mathrm {supp}} \widetilde {W}_{N,0} \subset A_R$
with
$A_R = \{ (x,v) \in \mathbb {R}^6 , \left | v \right | \leq R \}$
for some
$R>0$
because of Proposition 2.2. If in addition
$N^{-1} \mathcal {E}^{\mathrm {{MS}}}[\omega _{N,0}, \alpha _0] \leq C$
holds,
$\widetilde {C}(t)$
and
$C(t)$
can be bounded uniformly in N by a constant depending on time but not on the number of particles. Furthermore, the solution of (9) exists and is unique thanks to Proposition 2.1.
Remark 2.4. Next, we provide initial data fulfilling the assumptions of the theorem and such that
$\Xi (0)$
is small. Let us define
$f_{pr}(x) = \varepsilon ^{-3/2} e^{- i p \cdot x / \varepsilon } g(x-r)$
with
$g(x) = \frac {1}{(2 \pi \delta )^{3/4}} e^{- x^2 / (2 \delta ^2)}$
and
$\delta ^2 = \frac {\varepsilon }{\sqrt {2}}$
. Let
$M \in W_7^{8,2}(\mathbb {R}^6) \cap W_2^{0,1}(\mathbb {R}^6)$
such that
$0 \leq M(x,v) \leq 1$
,
$\int _{\mathbb {R}^6} M(x,v) \, dx \, dv = 1$
, and
$\text {supp} \{ M \} \subset \{ (x,v) \in \mathbb {R}^6 : |v| \leq R \} $
for some
$R>0$
. In addition, we define
$\eta _R: \mathbb {R}^3 \rightarrow \mathbb {R}$
by
and 
. Then,
$$ \begin{align*} \omega_{N,0}(x;y) &= \int_{\mathbb{R}^6} M(r,p) f_{p,r}(x) \overline{f_{pr}(y)} \, dp \, dr \quad \text{and} \quad \widetilde{W}_{N,0} = \eta_R \mathcal{W}[\omega_N] \end{align*} $$
satisfy the conditions of Theorem 2.1 and
$\Xi (0) \leq C \varepsilon $
. Note that the latter inequality can be shown by similar means as in [Reference Leopold33, Proof of Lemma III.7] using [Reference Leopold33, (37a) and (240)].
2.1 Comparison with the literature
To our knowledge the literature on the subject is rather limited. In [Reference Yang, Mauser and Möller59] Möller, Mauser and Yang considered the classical limit of the Pauli–Poisswell system, in which the coupling with the Maxwell equations is replaced by a simplified equation. For monokinetic initial data and by means of WKB methods, they show that as
$\hbar \to 0$
the system is approximated by the Euler-Poisswell equations. More recently, the massive Klein-Gordon-Maxwell system was derived from relativistic Euler-Maxwell system in the semi-classical limit in [Reference Salvi53] via an adapted modulated energy method. See also the related works [Reference Mauser and Selberg39, Reference Möller and Mauser41, Reference Möller40]. In [Reference Ben Porat46] the linear problem with external magnetic field has been considered. More precisely, the magnetic Liouville equation is obtained as classical limit of the Heisenberg equation with non constant magnetic field, whose vector potential is regular and given.
To the best of our knowledge, our result is the first derivation of the Vlasov–Maxwell system from the quantum dynamics described by the Maxwell–Schrödinger equations for extended charges, dealing with a self-consistent electromagnetic field which satisfies Maxwell’s equations. As outlined in Section 1.1, the Maxwell–Schrödinger equations can be heuristically obtained as mean-field limit of the spinless Pauli–Fierz Hamiltonian. A rigorous treatment of this derivation was provided in [Reference Leopold33], which appeared during the publication process of the current article. Moreover, by combining both results, a rigorous derivation of the Vlasov–Maxwell system from nonrelativistic quantum electrodynamics was established.
In this spirit, notice that a regularized variant of the relativistic Vlasov–Maxwell system as the mean-field limit of a system of many classical particles was derived in [Reference Golse25]. In the context of classical mechanics, we further refer to the works [Reference Lazarovici32, Reference Chen, Li, Pickl and Yin17], where the relativistic Vlasov–Maxwell system is obtained in a combined mean-field and point-particle limit of a system of N rigid charges with N-dependent radius.
Concerning the propagation of moments for the Vlasov–Maxwell system, we mention the works [Reference Rege49, Reference Rege50], dealing with a magnetized Vlasov–Poisson equation where the magnetic field is external and uniformly bounded.
We would also like to point out the works [Reference Spohn54, Reference Lions and Paul37, Reference Athanassoulis, Paul, Pezzotti and Pulvirenti7, Reference Amour, Khodja and Nourrigat2, Reference Golse, Mouhot and Paul26, Reference Benedikter, Porta, Saffirio and Schlein11, Reference Saffirio51, Reference Lafleche30, Reference Lewin and Sabin36, Reference Lafleche and Saffirio31, Reference Chen, Lee and Liew16, Reference Chong, Lafleche and Saffirio19] in which the related problem of the semiclassical approximation of the Hartree equation by the Vlasov system is addressed. We refer the interested reader to [Reference Saffirio52] for more details on the aforementioned papers.
2.2 Strategy of the proof
In order to prove Theorem 2.1 we adopt the approach of [Reference Benedikter, Porta, Saffirio and Schlein11] and compare the solutions of the Maxwell–Schrödinger equations (9) with the Weyl quantization (13) of the solutions of the Vlasov–Maxwell system (15) by means of a Grönwall estimate. The main novelty in comparison to [Reference Benedikter, Porta, Saffirio and Schlein11] is to deal with the additional difficulties arising from the coupling of the fermions to their self-induced electromagnetic field.
On the one hand a control on the difference of the vector potentials is required. This is achieved most efficiently by the mode functions
$\alpha _t$
and
$\widetilde {\alpha }_t$
of the electromagnetic fields, and gives the rationale for writing Maxwell’s equations in (9) and (15) in terms of
$\alpha _t$
and
$\widetilde {\alpha }_t$
.
On the other hand the fact that the vector potentials of the respective magnetic fields couple to the charge currents of the fermions forces us to measure the distance of the fermionic states in a Sobolev trace norm in which the Laplacian is weighted with the semiclassical parameter (see (35)).
Beside the semiclassical analysis, we deal with the well-posedness of the Maxwell–Schrödinger equations and the Vlasov–Maxwell equations and prove that there exist initial data whose evolution leads to the integrability and regularity conditions required by Theorem 2.1. This is shown in Proposition 2.1 and Proposition 2.2 by means of fixed point arguments – leading to local solutions – and suitable propagation estimates. Crucial technical ingredients of our approach to obtain the result for the Maxwell–Schrödinger equations are the use of propagation estimates for the time evolution of the magnetic Laplacian from [Reference Nakamura and Wada42] and the specific choice of the Banach space (171). In the case of the Vlasov–Maxwell system the compact support in the velocity of the initial data is the key assumption to overcome the difficulties arising from the interaction between the electromagnetic field and the nonrelativistic fermions with nonfinite speed of propagation. For point particles this assumption has already been used to prove well-posedness in different functional frameworks, for instance in [Reference Degond22].
3 Preliminaries
Within this section we collect important estimates that will often appear in the proofs of our results.
Lemma 3.1. Let the Assumption 2.1 hold. Then
$$ \begin{align} \left\| \eta * \boldsymbol{A}_{\alpha} \right\| _{L^{\infty}(\mathbb{R}^3)} &\leq C \min \left\{ \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{-1}}, \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{-1/2}}, \left\| \alpha \right\| _{\mathfrak{h}} , \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}} \right\} , \end{align} $$
$$ \begin{align} \left\| \eta * \boldsymbol{A}_{\alpha} \right\| _{L^{2}(\mathbb{R}^3)} &\leq C \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{-1/2}}. \end{align} $$
For
$\sigma \in \mathbb {N}$
we, moreover, have
$$ \begin{align} \left\| \eta * \boldsymbol{A}_{\alpha} \right\| _{W_0^{1, \infty}(\mathbb{R}^3)} &\leq C \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}} , \end{align} $$
$$ \begin{align} \left\| \eta * \boldsymbol{A}_{\alpha} \right\| _{W_0^{\sigma, \infty}(\mathbb{R}^3)} &\leq C \left\| \alpha \right\| _{\mathfrak{h}_{\sigma -1}}. \end{align} $$
Proof. Note that
$$ \begin{align} \left\| \eta * \boldsymbol{A} \right\| _{L^{\infty}(\mathbb{R}^3)} &\leq 2 \sum_{\lambda = 1,2} \int \left| k \right|{}^{-1/2} \left| \mathcal{F}[\eta](k) \right| \left| \alpha(k,\lambda) \right|dk \nonumber \\ &\leq C \min \Big\{ \left\| \left| \cdot \right|{}^{1/2} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \left| \cdot \right|{}^{-1} \alpha \right\| _{\mathfrak{h}} , \left\| \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \left| \cdot \right|{}^{-1/2} \alpha \right\| _{\mathfrak{h}} , \nonumber \\ &\qquad \qquad \qquad \left\| \left| \cdot \right|{}^{-1/2} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \alpha \right\| _{\mathfrak{h}} , \left\| \left| \cdot \right|{}^{-1} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \left| \cdot \right|{}^{1/2} \alpha \right\| _{\mathfrak{h}} \Big\}. \end{align} $$
Together with (28) this proves the first inequality. Similarly,
$$ \begin{align} \left\| \eta * \boldsymbol{A} \right\| _{W_0^{1,\infty}(\mathbb{R}^3)} &\leq 2 \sum_{\lambda = 1,2} \int \left| k \right|{}^{-1/2} \left< k \right> \left| \mathcal{F}[\eta](k) \right| \left| \alpha(k,\lambda) \right|dk \nonumber \\ &\leq C \left\| \left| \cdot \right|{}^{-1} \left< \cdot \right> \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \left| \cdot \right|{}^{1/2} \alpha \right\| _{\mathfrak{h}} \end{align} $$
implies (40a) and from
$$ \begin{align} \left\| \eta * \boldsymbol{A} \right\| _{W_0^{\sigma,\infty}(\mathbb{R}^3)} &\leq 2 \sum_{\lambda = 1,2} \int \left| k \right|{}^{-1/2} \left< k \right>^{\sigma} \left| \mathcal{F}[\eta](k) \right| \left| \alpha(k,\lambda) \right|dk \nonumber \\ &\leq C \left\| \left| \cdot \right|{}^{-1/2} \left< \cdot \right> \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \alpha \right\| _{\mathfrak{h}_{\sigma - 1}} \end{align} $$
we get (40b). By means of
$$ \begin{align} \boldsymbol{A}_{\alpha}(x) = (2 \pi )^{- 3/2} \sum_{\lambda = 1,2} \int \frac{1}{\sqrt{2 \left| k \right|}} \boldsymbol{\epsilon}_{\lambda}(k) \left( e^{i k x} \alpha(k,\lambda) + e^{- i k x} \overline{\alpha(k, \lambda)} \right)\,dk\,, \end{align} $$
Young’s inequality and (27) we have
$$ \begin{align} \left\| \eta * \boldsymbol{A}_{\alpha} \right\| _{L^{2}(\mathbb{R}^3)} &\leq \left\| \eta \right\| _{L^1(\mathbb{R}^3)} \left\| \boldsymbol{A}_{\alpha} \right\| _{L^2(\mathbb{R}^3)} \leq C \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{-1/2}}. \end{align} $$
Moreover, the following estimates will be useful
Lemma 3.2. Let
$s \in \{ 0,1,2 \}$
and
$\sigma \in \mathbb {N}$
. Then,
$$ \begin{align} \left\| K * \rho_{\omega} \right\| _{W_0^{3, \infty}(\mathbb{R}^3)} &\leq C N^{-1} \left\| \omega \right\| _{\mathfrak{S}^1} \quad \text{for} \quad \omega \in \mathfrak{S}^1(L^2(\mathbb{R}^3)) , \end{align} $$
$$ \begin{align} \left\| K * \widetilde{\rho}_{f} \right\| _{W_0^{\sigma ,\infty}(\mathbb{R}^3)} &\leq C \begin{cases} \min \left\{ \left\| f \right\| _{W_4^{0,2}(\mathbb{R}^6)} , \left\| f \right\| _{L^1(\mathbb{R}^6)} \right\} &\text{if} \quad \sigma \leq 3 , \\ \left\| f \right\| _{W_4^{\sigma - 3,2}(\mathbb{R}^6)} &\text{if} \quad \left| \sigma \right|> 3 \end{cases} \end{align} $$
and
$$ \begin{align} \left\| X_{\omega} \right\| _{\mathfrak{S}^{\infty}(L^2(\mathbb{R}^3))} &\leq C N^{-1} \left\| \omega \right\| _{\mathfrak{S}^1} \quad \text{for} \quad \omega \in \mathfrak{S}^1(L^2(\mathbb{R}^3)) , \end{align} $$
$$ \begin{align} \left\| \left[ X_{\omega} , \Gamma \right] \right\| _{\mathfrak{S}^{s,1}} &\leq C N^{-1} \left\| \omega \right\| _{\mathfrak{S}^{s,1}} \left\| \Gamma \right\| _{\mathfrak{S}^{s,1}} \quad \text{for} \quad \omega, \Gamma \in \mathfrak{S}^{s,1}(L^2(\mathbb{R}^3)). \end{align} $$
Proof. Due to (10a) we have
$$ \begin{align} \mathcal{F}[K](k) &= \mathcal{F}[\eta * \eta * \left| \cdot \right|{}^{-1}](k) = C \mathcal{F}[\eta](k)^2 \left| k \right|{}^{-2}. \end{align} $$
Together with Young’s inequality and (28) this implies
$$ \begin{align} \left\| K * \rho_{\omega} \right\| _{W_0^{\sigma, \infty}(\mathbb{R}^3)} &\leq C \int \left< k \right>^{\sigma} |\mathcal{F}[K](k)||\hat{\rho}_\omega(k)|\,dk \nonumber \\ &\leq \left\| \left< \cdot \right>^3 \mathcal{F}[K] \right\| _{L^1(\mathbb{R}^3)} \left\| (1 - \Delta)^{(\sigma - 3)/2} \rho_{\omega} \right\| _{L^1(\mathbb{R}^3)} \nonumber \\ &\leq C \left\| \left< \cdot \right>^{3/2} \left| \cdot \right|{}^{-1} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)}^2 \left\| (1 - \Delta)^{(\sigma - 3)/2} \rho_{\omega} \right\| _{L^1(\mathbb{R}^3)} \nonumber \\ &\leq C \left\| (1 - \Delta)^{(\sigma - 3)/2} \rho_{\omega} \right\| _{L^1(\mathbb{R}^3)}. \end{align} $$
Inequality (46a) is then obtained by setting
$\sigma = 3$
and
$$ \begin{align} \left\| \rho_{\omega} \right\| _{L^1(\mathbb{R}^3)} &= \sup_{O \in L^{\infty}(\mathbb{R}^3), \left\| O \right\| _{L^{\infty}(\mathbb{R}^3)} \leq 1} \left| \int dx \, O(x) \rho_{\omega}(x) \right| \leq N^{-1} \left\| \omega \right\| _{\mathfrak{S}^1}, \end{align} $$
which holds because the space of bounded operators is the dual space of trace-class operators and every function
$x \rightarrow O(x)$
defines a multiplication operator. Next, we replace
$\rho _{\omega }$
in (49) by
$\widetilde {\rho }_f$
. The case
$\sigma \leq 3$
in (46b) is an immediate consequence of (16a). In order to obtain the estimate for
$\sigma> 3$
we bound the last line of (49) with
$\rho _{\omega } = \widetilde {\rho }_f$
by
$$ \begin{align} \left\| (1 - \Delta)^{(\sigma - 3)/2} \widetilde{\rho}_{f} \right\| _{L^{1}(\mathbb{R}^3)} &\leq \int_{\mathbb{R}^6} \left< (x,v) \right>^{-4} \left< (x,v) \right>^4 \left| (1 - \Delta_x)^{(\sigma - 3)/2} f(x,v) \right| \, dx \, dv \nonumber \\ &\leq C \left\| \left< \cdot \right>^4 (1 - \Delta)^{(\sigma - 3)/2} f \right\| _{L^2(\mathbb{R}^6)} \nonumber \\ &\leq \left\| f \right\| _{W_4^{\sigma-3,2}(\mathbb{R}^6)}. \end{align} $$
Using the Fourier decomposition
$$ \begin{align} K(x-y) &= \frac{1}{(2 \pi)^{3/2}} \int e^{i k (x-y)} \mathcal{F}[K](k)\,dk \end{align} $$
we write the exchange term (with
$\hat {x}$
denoting the operator of multiplication by x) as
$$ \begin{align} X_{\omega} &= \frac{1}{(2 \pi)^{3/2} N} \int \mathcal{F}[K](k) e^{i k \hat{x}} \omega e^{- i k \hat{x}}dk \end{align} $$
and estimate
$$ \begin{align} \left\| X_{\omega} \right\| _{\mathfrak{S}^{\infty}} &\leq N^{-1} \left\| \mathcal{F}[K] \right\| _{L^1(\mathbb{R}^3)} \left\| \omega \right\| _{\mathfrak{S}^{\infty}}. \end{align} $$
By applying the bound
$\left \| \mathcal {F}[K] \right \| _{L^1(\mathbb {R}^3)} \leq C \left \| \left | \cdot \right |{}^{-1} \mathcal {F}[\eta ] \right \| _{L^2(\mathbb {R}^3)}^2 \leq C$
(which follows from (48) and (28)) together with the inequality
$\left \| \omega \right \| _{\mathfrak {S}^{\infty }} \leq \left \| \omega \right \| _{\mathfrak {S}^{1}}$
we obtain
$$ \begin{align} \left\| X_{\omega} \right\| _{\mathfrak{S}^{\infty}} \leq C N^{-1} \left\| \omega \right\| _{\mathfrak{S}^{\infty}} \leq C N^{-1} \left\| \omega \right\| _{\mathfrak{S}^1} \end{align} $$
and
$$ \begin{align} \left\| \left[ X_{\omega} , \Gamma \right] \right\| _{\mathfrak{S}^1} &\leq 2 \left\| X_{\omega} \right\| _{\mathfrak{S}^{\infty}(L^2(\mathbb{R}^3))} \left\| \Gamma \right\| _{\mathfrak{S}^1} \leq C N^{-1} \left\| \omega \right\| _{\mathfrak{S}^1} \left\| \Gamma \right\| _{\mathfrak{S}^1}. \end{align} $$
In the following let
$s \in \{1,2 \}$
. If
$\omega \in \mathfrak {S}^{s,1}(L^2(\mathbb {R}^3))$
we can split the compact and self-adjoint operator
$(1 - \Delta )^{s/2} \omega (1 - \Delta )^{s/2} = \left ( (1 - \Delta )^{s/2} \omega (1 - \Delta )^{s/2} \right )_{+} - \left ( (1 - \Delta )^{s/2} \omega (1 - \Delta )^{s/2} \right )_{-}$
into its positive and negative part. Then
$$ \begin{align} \left( (1 - \Delta)^{s/2} \omega (1 - \Delta)^{s/2} \right)_{+} \left( (1 - \Delta)^{s/2} \omega (1 - \Delta)^{s/2} \right)_{-} &= 0 \end{align} $$
and
$$ \begin{align} \left| (1 - \Delta)^{s/2} \omega (1 - \Delta)^{s/2} \right| &= \left( (1 - \Delta)^{s/2} \omega (1 - \Delta)^{s/2} \right)_{+} + \left( (1 - \Delta)^{s/2} \omega (1 - \Delta)^{s/2} \right)_{-}. \end{align} $$
We define the positive operators
$$ \begin{align} \omega_{+} &= (1 - \Delta)^{- s/2} \left( (1 - \Delta)^{s/2} \omega (1 - \Delta)^{s/2} \right)_{+} (1 - \Delta)^{-s/2} , \nonumber \\ \omega_{-} &= (1 - \Delta)^{-s/2} \left( (1 - \Delta)^{s/2} \omega (1 - \Delta)^{s/2} \right)_{-} (1 - \Delta)^{-s/2} \end{align} $$
which satisfy
$$ \begin{align} \omega_{+} - \omega_{-} = \omega \quad \text{and} \quad \left\| \omega \right\| _{\mathfrak{S}^{s,1}} = \left\| (1 - \Delta)^{s/2} \omega (1 - \Delta)^{s/2} \right\| _{\mathfrak{S}^{1}} = \left\| \omega_{+} \right\| _{\mathfrak{S}^{s,1}} + \left\| \omega_{-} \right\| _{\mathfrak{S}^{s,1}}. \end{align} $$
Note that
$\omega _{+/-} \in \mathfrak {S}^{s,1}(L^2(\mathbb {R}^3))$
because
$\omega \in \mathfrak {S}^{s,1}(L^2(\mathbb {R}^3))$
. This splitting and the linearity of the mapping
$\omega \mapsto X_{\omega }$
yield
$$ \begin{align} \left\| \left[ X_{\omega} , \Gamma \right] \right\| _{\mathfrak{S}^{s,1}} &\leq 2 \left\| \left( 1 - \Delta \right)^{s/2} X_{\omega_+} (1 - \Delta)^{-s/2} \right\| _{\mathfrak{S}^{\infty}(L^2(\mathbb{R}^3))} \left\| \Gamma \right\| _{\mathfrak{S}^{s,1}} \nonumber \\ &\quad + 2 \left\| \left( 1 - \Delta \right)^{s/2} X_{\omega_-} (1 - \Delta)^{-s/2} \right\| _{\mathfrak{S}^{\infty}(L^2(\mathbb{R}^3))} \left\| \Gamma \right\| _{\mathfrak{S}^{s,1}}. \end{align} $$
Since
$\omega _{+} \in \mathfrak {S}^{s,1}(L^2(\mathbb {R}^3))$
there exists a spectral set
$\{ \lambda _j, \varphi _j \}_{j \in \mathbb {N}}$
with
$\lambda _j> 0$
for all
$j \in \mathbb {N}$
such that
$\omega _{+} = \sum _{j \in \mathbb {N}} \lambda _j | \varphi _j \rangle \langle \varphi _j |$
. Because
$(1 - \Delta )^{s/2} \varphi _j = \lambda _j^{-1} (1 - \Delta )^{s/2} \omega _{+} \varphi _j$
we get
$$ \begin{align} \left\| \varphi_j \right\| _{H^s(\mathbb{R}^3)} &\leq \lambda_j^{-1} \left\| (1 - \Delta)^{s/2} \omega_{+} (1 - \Delta)^{s/2} \right\| _{\mathfrak{S}^{\infty}(L^2(\mathbb{R}^3))} \left\| (1 - \Delta)^{-s/2} \varphi_j \right\| _{L^2(\mathbb{R}^3)} \nonumber \\ &\leq \lambda_j^{-1} \left\| (1 - \Delta)^{s/2} \omega_{+} (1 - \Delta)^{s/2} \right\| _{\mathfrak{S}^1(L^2(\mathbb{R}^3))} \left\| \varphi_j \right\| _{L^2(\mathbb{R}^3)}. \end{align} $$
We consequently have
$\varphi _j \in H^s(\mathbb {R}^3)$
,
$$ \begin{align} \left( 1 - \Delta \right)^{s/2} \omega_{+} \left( 1 - \Delta \right)^{s/2} = \sum_{j \in \mathbb{N}} \lambda_j | \left( 1 - \Delta \right)^{s/2} \varphi_j \rangle \langle \left( 1 - \Delta \right)^{s/2} \varphi_j | \end{align} $$
and
$$ \begin{align} \left\| \omega_{+} \right\| _{\mathfrak{S}^{s,1}} &= \text{Tr} \left( \left( 1 - \Delta \right)^{s/2} \omega_{+} \left( 1 - \Delta \right)^{s/2} \right) = \sum_{j \in \mathbb{N}} \lambda_j \left\| \varphi_j \right\| _{H^s(\mathbb{R}^3)}^2. \end{align} $$
Using this spectral decomposition, the product rule of differentiation, Young’s inequality and the Cauchy–Schwarz inequality, we obtain (
$\chi = (1 - \Delta )^{- s/2} \psi $
)
$$ \begin{align} \left\| \left( 1 - \Delta \right)^{s/2} X_{\omega_{+}} \left( 1 - \Delta \right)^{-s/2} \psi \right\| _{L^2(\mathbb{R}^3)} & = N^{-1} \Big\| \sum_{j \in \mathbb{N}} \lambda_j \varphi_j K * \{ \overline{\varphi_j} \chi \} \Big\|_{H^s(\mathbb{R}^3)} \nonumber \\ &\leq C N^{-1} \left\| \chi \right\| _{L^2(\mathbb{R}^3)} \left\| \left< \cdot \right>^s \mathcal{F}[K] \right\| _{L^1(\mathbb{R}^3)} \sum_{j \in \mathbb{N}} \lambda_j \left\| \varphi_j \right\| _{H^s(\mathbb{R}^3)}^2 \nonumber \\ &\leq C N^{-1} \left\| \psi \right\| _{L^2(\mathbb{R}^3)} \left\| \left< \cdot \right>^s \mathcal{F}[K] \right\| _{L^1(\mathbb{R}^3)} \left\| \omega_{+} \right\| _{\mathfrak{S}^{s,1}}. \end{align} $$
Similarly as above, (48) and (28) yield
$\left \| \left < \cdot \right>^s \mathcal {F}[K] \right \| _{L^1(\mathbb {R}^3)} \leq \left \| \left < \cdot \right>^2 \mathcal {F}[K] \right \| _{L^1(\mathbb {R}^3)} \leq C$
. This shows
$$ \begin{align} &\left\| \left( 1 - \Delta \right)^{s/2} X_{\omega_+} (1 - \Delta)^{- s/2} \right\| _{\mathfrak{S}^{\infty}(L^2(\mathbb{R}^3))} \nonumber \\ &\quad = \sup_{\psi \in L^2(\mathbb{R}^3), \left\| \psi \right\| _{L^2(\mathbb{R}^3)} = 1} \left\| \left( 1 - \Delta \right)^{s/2} X_{\omega_{+}} \left( 1 - \Delta \right)^{- s/2} \psi \right\| _{L^2(\mathbb{R}^3)} \nonumber \\ &\quad \leq C N^{-1} \left\| \omega_{+} \right\| _{\mathfrak{S}^{s,1}}. \end{align} $$
Analogously we obtain the same estimate for
$X_{\omega _{-}}$
, that is,
$$ \begin{align} \left\| \left( 1 - \Delta \right)^{s/2} X_{\omega_{-}} (1 - \Delta)^{- s/2} \right\| _{\mathfrak{S}^{\infty}(L^2(\mathbb{R}^3))} &\leq C N^{-1} \left\| \omega_{-} \right\| _{\mathfrak{S}^{s,1}}. \end{align} $$
In total, this shows
$\left \| \left [ X_{\omega }, \Gamma \right ] \right \| _{\mathfrak {S}^{s,1}} \leq C N^{-1} \left \| \omega \right \| _{\mathfrak {S}^{s,1}} \left \| \Gamma \right \| _{\mathfrak {S}^{s,1}} $
for
$s \in \{ 1,2 \}$
.
Lemma 3.3. Let
$\boldsymbol {J}$
be defined as in (10d) and
$\left \| \omega \right \| _{\mathfrak {S}_{\varepsilon }^{1,1}} = \left \| (1 - \varepsilon ^2 \Delta )^{1/2} \omega (1 - \varepsilon ^2 \Delta )^{1/2} \right \| _{\mathfrak {S}^1}$
. Then,
$$ \begin{align} &\left\| \mathcal{F}[\boldsymbol{J}_{\omega,\alpha}] - \mathcal{F}[\boldsymbol{J}_{\omega',\alpha'}] \right\| _{L^{\infty}(\mathbb{R}^3)} \nonumber \\ &\quad\leq C \left( 1 + \min \left\{ \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{-1/2}} , \left\| \alpha \right\| _{\mathfrak{h}}, \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}} \right\} \right) N^{-1} \left\| \omega - \omega' \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} \nonumber \\ &\qquad + C N^{-1} \left\| \omega' \right\| _{\mathfrak{S}^1} \min \left\{ \left\| \alpha - \alpha' \right\| _{\dot{\mathfrak{h}}_{-1/2}} , \left\| \alpha - \alpha' \right\| _{\mathfrak{h}}, \left\| \alpha - \alpha' \right\| _{\dot{\mathfrak{h}}_{1/2}} \right\}. \end{align} $$
For
$\mathcal {E}^{\mathrm {{MS}}}$
being defined as in (31) and
$(\omega ,\alpha ) \in \mathfrak {S}_{+}^{1,1} \times \dot {\mathfrak {h}}_{1/2}$
we have
$$ \begin{align} \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 &\leq N^{-1} \mathcal{E}^{\mathrm{{MS}}}[\omega, \alpha] + C N^{-2} \left\| \omega \right\| _{\mathfrak{S}^1}^2 , \end{align} $$
$$ \begin{align} \text{Tr} \left( - \varepsilon^2 \Delta \omega \right) &\leq C \left( \mathcal{E}^{\mathrm{{MS}}}[\omega, \alpha] + N^{-1} \left\| \omega \right\| _{\mathfrak{S}^1}^2 \right) \left( 1 + N^{-1} \left\| \omega \right\| _{\mathfrak{S}^1} \right) , \end{align} $$
$$ \begin{align} \mathcal{E}^{\mathrm{{MS}}}[\omega, \alpha] &\leq \text{Tr} \left( - \varepsilon^2 \Delta \omega \right) + C N^{-1} \left\| \omega \right\| _{\mathfrak{S}^1}^2 + \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 \left( C \left\| \omega \right\| _{\mathfrak{S}^1} + N \right). \end{align} $$
Proof. We write the difference of the currents as
$$ \begin{align} \boldsymbol{J}_{\omega, \alpha}(x) - \boldsymbol{J}_{\omega', \alpha'}(x) &= - N^{-1} \left\{ i \varepsilon \nabla, \left( \omega - \omega' \right) \right\}(x;x) - 2 \rho_{\omega - \omega'}(x) \eta * \boldsymbol{A}_{\alpha}(x) \nonumber \\ &\quad - 2 \rho_{\omega'}(x) \eta * \boldsymbol{A}_{\alpha - \alpha'}(x). \end{align} $$
The Fourier transform of N times the first expression on the right-hand side can be estimated by
$$ \begin{align} \left\| \mathcal{F} [\left\{ i \varepsilon \nabla, \left( \omega_{N,s} - \widetilde{\omega}_{N,s} \right) \right\}(\cdot; \cdot)] \right\| _{L^{\infty}(\mathbb{R}^3)} &\leq \sup_{k \in \mathbb{R}^3} \left| \int dx \, e^{- i k x} \left\{ i \varepsilon \nabla, \left( \omega_{N,s} - \widetilde{\omega}_{N,s} \right) \right\}(x;x) \right| \nonumber \\ &= \sup_{k \in \mathbb{R}^3} \left| \text{Tr} \left( e^{- i k \hat{x}} \left\{ i \varepsilon \nabla, \left( \omega_{N,s} - \widetilde{\omega}_{N,s} \right) \right\} \right) \right| \nonumber \\ &\leq \left\| i \varepsilon \nabla \left( \omega_{N,s} - \widetilde{\omega}_{N,s} \right) \right\| _{\mathfrak{S}^1} + \left\| \left( \omega_{N,s} - \widetilde{\omega}_{N,s} \right) i \varepsilon \nabla \right\| _{\mathfrak{S}^1} \nonumber \\ &\leq 2 \left\| \left( \omega_{N,s} - \widetilde{\omega}_{N,s} \right) \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}}. \end{align} $$
To obtain the ultimate expression we have used that
$\left \| i \varepsilon \nabla \left ( 1 - \varepsilon \Delta \right )^{-1/2} \right \| _{\mathfrak {S}^{\infty }} \leq 1$
. Using (39a) and (50) we bound the second and third term by
$$ \begin{align} 2 \left\| \mathcal{F}[ \rho_{\omega - \omega'} \, \eta * \boldsymbol{A}_{\alpha} ] \right\| _{L^{\infty}(\mathbb{R}^3)} &\leq \left\| \rho_{\omega - \omega'} \, \eta * \boldsymbol{A}_{\alpha} \right\| _{L^1(\mathbb{R}^3)} \nonumber \\ &\leq \left\| \eta * \boldsymbol{A}_{\alpha} \right\| _{L^{\infty}(\mathbb{R}^3)} \left\| \rho_{\omega - \omega'} \right\| _{L^1(\mathbb{R}^3)} \nonumber \\ &\leq C N^{-1} \min \left\{ \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{-1/2}} , \left\| \alpha \right\| _{\mathfrak{h}}, \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}} \right\} \left\| \omega - \omega' \right\| _{\mathfrak{S}^1} \end{align} $$
and
$$ \begin{align} 2 \left\| \mathcal{F}[ \rho_{\omega'} \, \eta * \boldsymbol{A}_{\alpha - \alpha'}] \right\| _{L^{\infty}(\mathbb{R}^3)} &\leq C N^{-1} \min \left\{ \left\| \alpha - \alpha' \right\| _{\dot{\mathfrak{h}}_{-1/2}} , \left\| \alpha - \alpha' \right\| _{\mathfrak{h}}, \left\| \alpha - \alpha' \right\| _{\dot{\mathfrak{h}}_{1/2}} \right\} \left\| \omega' \right\| _{\mathfrak{S}^1}. \end{align} $$
Collecting the estimates proves (68). Inequality (69a) follows from the positivity of the magnetic Laplacian and Lemma 3.2. Using
$(- i \varepsilon \nabla - \eta * \boldsymbol {A}_{\alpha })^2 \geq - \varepsilon ^2 \Delta + 2 i \varepsilon \nabla \cdot \eta * \boldsymbol {A}_{\alpha }$
,
$\left \| i \varepsilon \nabla \omega \right \| _{\mathfrak {S}^1}^2 \leq \text {Tr} \left ( - \varepsilon ^2 \Delta \omega \right ) \left \| \omega \right \| _{\mathfrak {S}^1}$
if
$\omega \geq 0$
, (39a) and Lemma 3.2 we get
$$ \begin{align} \text{Tr} \left( - \varepsilon^2 \Delta \omega \right) &\leq 2 \text{Tr} \left( (- i \varepsilon \nabla - \eta * \boldsymbol{A}_{\alpha})^2 \omega \right) + C \left\| \omega \right\| _{\mathfrak{S}^1} \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 \nonumber \\ &\leq 2 \mathcal{E}^{\mathrm{{MS}}}[\omega, \alpha] + C \left\| \omega \right\| _{\mathfrak{S}^1} \left( \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 + N^{-1} \left\| \omega \right\| _{\mathfrak{S}^1} \right). \end{align} $$
Together with (69a) this leads to (69b). By similar estimates we obtain inequality (69c) concluding the proof.
Lemma 3.4. Let
$(f, \alpha ) \in W_{2}^{0,1}(\mathbb {R}^6) \times \dot {\mathfrak {h}}_{1/2}$
such that
$f \geq 0$
. For
$\mathcal {E}^{\mathrm {{VM}}}$
,
$\boldsymbol {F}_{f,\alpha }$
and
$\boldsymbol {J}_{f, \alpha }$
defined as in (33), (16b) and (16c) we have
$$ \begin{align} \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 &\leq \mathcal{E}^{\mathrm{{VM}}}[f,\alpha] + C \left\| f \right\| _{L^1(\mathbb{R}^6)}^2 , \end{align} $$
$$ \begin{align} \int_{\mathbb{R}^6} f(x,v) v^2dx\,dv &\leq \mathcal{E}^{\mathrm{{VM}}}[f,\alpha] + C \left\| f \right\| _{L^1(\mathbb{R}^6)} \left( \left\| f \right\| _{L^1(\mathbb{R}^6)} + \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 \right) , \end{align} $$
$$ \begin{align} \mathcal{E}^{\mathrm{{VM}}}[f,\alpha] &\leq 2 \int_{\mathbb{R}^6} f(x,v) v^2dx\,dv + C \left\| f \right\| _{L^1(\mathbb{R}^6)}^2 + \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 \left( C \left\| f \right\| _{L^1(\mathbb{R}^6)} + 1 \right) \end{align} $$
and
$$ \begin{align} \sup_{x \in \mathbb{R}^3} \left| \boldsymbol{F}_{f, \alpha}(x,v) \right| &\leq C \left< v \right> \left( 1 + \mathcal{E}^{\mathrm{{VM}}}[f, \alpha] + C \left\| f \right\| _{L^1(\mathbb{R}^6)}^2 \right) , \end{align} $$
$$ \begin{align} \left\| \widetilde{\boldsymbol{J}}_{f,\alpha} \right\| _{L^1(\mathbb{R}^3)} &\leq C \left( 1 + \mathcal{E}^{\mathrm{{VM}}}[f, \alpha] + C \left\| f \right\| _{L^1(\mathbb{R}^6)}^2 \right) , \end{align} $$
$$ \begin{align} \left\| \widetilde{\boldsymbol{J}}_{f,\alpha} \right\| _{W_0^{\sigma,1}(\mathbb{R}^3)} &\leq C \left( 1 + \left\| \alpha \right\| _{\mathfrak{h}_{\sigma -1}} \right) \left\| f \right\| _{W_5^{\sigma,2}(\mathbb{R}^6)} \quad \text{with} \; \sigma \in \mathbb{N}_0. \end{align} $$
Proof. Note that
$$ \begin{align} \left| \int_{\mathbb{R}^6} f(x,v) \left( \eta * \boldsymbol{A}_{\alpha}(x) \right)^2dx\,dv \right| &\leq C \left\| f \right\| _{L^1(\mathbb{R}^6)} \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 \end{align} $$
and
$$ \begin{align} \left| \int_{\mathbb{R}^6} f(x,v) K * \widetilde{\rho}_f(x)\,dx\,dv \right| &\leq C \left\| f \right\| _{L^1(\mathbb{R}^6)}^2 \end{align} $$
because of (39a) and (46b). The inequalities (75a)–(75c) then follow from the positivity of f,
$(v - \eta * \boldsymbol {A}_{\alpha }(x))^2 \geq v^2 - 2 v \cdot \eta \boldsymbol {A}_{\alpha }(x)$
and the Cauchy–Schwarz inequality. By means of (40a) and (46b) and the Cauchy–Schwarz inequality we obtain
$$ \begin{align} \sup_{x \in \mathbb{R}^3} \left| \boldsymbol{F}_{f, \alpha}(x,v) \right| &\leq C \left< v \right> \left( \left\| f \right\| _{L^1(\mathbb{R}^6)} + \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 \right) \end{align} $$
and
$$ \begin{align} \left\| \widetilde{\boldsymbol{J}}_{f,\alpha} \right\| _{L^1(\mathbb{R}^3)} &\leq C \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}} \left\| f \right\| _{L^1(\mathbb{R}^6)} + 2 \left( \left\| f \right\| _{L^1(\mathbb{R}^6)} \int_{\mathbb{R}^6} f(x,v) v^2dx\,dv \right)^{1/2}. \end{align} $$
Together with (75a) and (75b) this leads to (76a) and (76b). Note that
$$ \begin{align} \left\| \widetilde{\boldsymbol{J}}_{f,\alpha} \right\| _{W_0^{\sigma,1}(\mathbb{R}^3)} &\leq 2 \left( 1 + \left\| \eta * \boldsymbol{A}_{\alpha} \right\| _{W_0^{\sigma, \infty}(\mathbb{R}^3)} \right) \left\| f \right\| _{W_1^{\sigma,1}}. \end{align} $$
By applying (40b) and the Cauchy-Schwarz inequality we then obtain (76c).
Lemma 3.5. Let A be a trace class operator,
$V: \mathbb {R}^3 \rightarrow \mathbb {C}$
be a regular enough function and
$D_{\leq n} = \frac {\left ( 1 - \varepsilon ^2 \Delta \right )^{1/2}}{\left ( 1 - \varepsilon ^2 \Delta / n^2 \right )^{1/2}} $
with
$n \in \mathbb {N}$
as defined in (100). Then,
$$ \begin{align} \left\| \left[ V, A \right] \right\| _{\mathfrak{S}^1} &\leq \left\| \left[ \hat{x}, A \right] \right\| _{\mathfrak{S}^1} \int \left| k \right| \left| \mathcal{F}[V](k) \right|\,dk , \end{align} $$
$$ \begin{align} \left\| \left[ \left( 1 - \varepsilon^2 \Delta \right)^{1/2} , V \right] \right\| _{\mathfrak{S}^{\infty}} &\leq \varepsilon \int \left| k \right| \left| \mathcal{F}[V](k) \right|\,dk , \end{align} $$
$$ \begin{align} \left\| \left[ D_{\leq n} , V \right] \right\| _{\mathfrak{S}^{\infty}} &\leq \varepsilon \int \left| k \right| \left| \mathcal{F}[V](k) \right|\,dk. \end{align} $$
Proof. Using the identity
$$ \begin{align} \left[ e^{i k \hat{x}} , A \right] &= \int_0^1 \, e^{i \lambda k \hat{x}} \left[ i k \cdot \hat{x} , A \right] e^{i (1 - \lambda) k \hat{x}}\,d \lambda, \end{align} $$
we estimate
$$ \begin{align} \left\| \left[ V, A \right] \right\| _{\mathfrak{S}^1} &\leq (2 \pi)^{-3/2} \int \left| \mathcal{F}[V](k) \right| \left\| \left[ e^{ik \hat{x}} , A \right] \right\| _{\mathfrak{S}^1}dk \nonumber \\ &\leq \int\left| k \right| \left| \mathcal{F}[V](k) \right| \left\| \left[ \hat{x} , A \right] \right\| _{\mathfrak{S}^1}dk. \end{align} $$
Note that
$$ \begin{align} (2 \pi)^{3/2} \left[ \left( 1 - \varepsilon^2 \Delta \right)^{1/2} , V(x) \right] &= \int \mathcal{F}[V](k) \left[ \left( 1 - \varepsilon^2 \Delta \right)^{1/2} , e^{ikx} \right]\,dk \nonumber \\ &= \int \mathcal{F}[V](k) e^{ikx} \left( \left( 1 + \varepsilon^2 (i \nabla - k)^2 \right)^{1/2} - \left( 1 - \varepsilon^2 \Delta \right)^{1/2} \right)\,dk \nonumber \\ &= - \int \mathcal{F}[V](k) e^{ikx} \frac{ \varepsilon^2 k \cdot i \nabla + \varepsilon^2 k \cdot (i \nabla - k)}{\left( \left( 1 + \varepsilon^2 (i \nabla - k)^2 \right)^{1/2} + \left( 1 - \varepsilon^2 \Delta \right)^{1/2} \right)}\,dk. \end{align} $$
By means of the operator inequalities
$\varepsilon (i \nabla - k) \leq \left ( 1 + \varepsilon ^2 (i \nabla - k)^2 \right )^{1/2}$
and
$\varepsilon i \nabla \leq \left ( 1 - \varepsilon ^2 \Delta \right )^{1/2}$
, which hold due to the spectral theorem, we obtain
$$ \begin{align} \left\| \left[ \left( 1 - \varepsilon^2 \Delta \right)^{1/2} , V(x) \right] \right\| _{\mathfrak{S}^{\infty}} &\leq \varepsilon \int \left| k \right| \left| \mathcal{F}[V](k) \right|\,dk. \end{align} $$
Inequality (82c) can be proven in a similar way.
4 Proof of Theorem 2.1
4.1 Quantization of the Vlasov–Maxwell equations
Lemma 4.1. Let
$(\widetilde {W}_{N,t}, \widetilde {\alpha }_t )$
be the solution of (15) and
$\widetilde {\omega }_{N,t}$
be the Weyl quantization of
$\widetilde {W}_{N,t}$
. Then
$(\widetilde {\omega }_{N,t} , \widetilde {\alpha }_t )$
satisfies
$$ \begin{align} \begin{cases} i \varepsilon \partial_t \widetilde{\omega}_{N,t} \kern-7pt&= \left[ - \varepsilon^2 \Delta, \widetilde{\omega}_{N,t} \right] + B_{t} + C_t, \\ \partial_t \widetilde{\alpha}_t(k,\lambda) \kern-7pt&= \left| k \right| \alpha_t(k,\lambda) - \sqrt{\frac{4 \pi^3}{\left| k \right|}} \mathcal{F}[\eta](k) \boldsymbol{\epsilon}_{\lambda}(k) \cdot \mathcal{F}[\boldsymbol{J}_{\widetilde{\omega}_{N,t}, \widetilde{\alpha}_t}](k) \end{cases} \end{align} $$
with
$$ \begin{align} B_t(x;y) &= \nabla (K * \rho_{\widetilde{\omega}_{N,t}}) \left( \frac{x + y}{2} \right) \cdot (x - y) \, \widetilde{\omega}_{N,t}(x;y) , \end{align} $$
$$ \begin{align} C_t(x;y) &= 2 \eta * \boldsymbol{A}_{\widetilde{\alpha}_t} \left( \frac{x + y}{2} \right) \cdot \left[ i \varepsilon \nabla, \widetilde{\omega}_{N,t} \right](x;y) + \nabla (\eta * \boldsymbol{A}_{\widetilde{\alpha}_t})^2 \left( \frac{x+y}{2} \right) \cdot (x - y) \, \widetilde{\omega}_{N,t}(x;y) \nonumber \\ &\quad + \sum_{j=1}^3 \nabla (\eta * \boldsymbol{A}_{\widetilde{\alpha}_t}^{(j)}) \left( \frac{x + y}{2} \right) \cdot (x - y) \left\{ i \varepsilon \nabla^{(j)} , \widetilde{\omega}_{N,t} \right\}(x;y). \end{align} $$
Here, K,
$\boldsymbol {A}$
,
$\rho $
and
$\boldsymbol {J}$
are defined as in (10a)–(10d).
Proof. To simplify the notation we refrain from writing
$\sim $
on top of the symbols and write
$W_{N,t}, \omega _{N,t}, \alpha _t, \ldots $
instead of
$\widetilde {W}_{N,t}, \widetilde {\omega }_{N,t}, \widetilde {\alpha }_t, \ldots $
. Using that
$W_{N}(t)$
satisfies the first equation in (15) and
$$ \begin{align} \tilde{\rho}_{W_{N,t}}(x) = \int W_{N,t}(x,v)\,dv = N^{-1} \omega_{N,t}(x,x) = \rho_{\omega_{N,t}}(x) \end{align} $$
we obtain
$$ \begin{align} &\partial_t \omega_{N,t}(x;y) \nonumber \\ &\quad = - N \int e^{i v \cdot \frac{x-y}{\varepsilon}} 2 \left( v - \eta * \boldsymbol{A}_{\alpha_t} \left( \frac{x + y}{2} \right) \right) \cdot \nabla_2 W_{N,t} \left( \frac{x+y}{2}, v \right)\,dv \end{align} $$
$$ \begin{align} &\qquad + N \int e^{i v \cdot \frac{x-y}{\varepsilon}} \, (\nabla K * \rho_{\omega_t}) \left( \frac{x+y}{2} \right) \cdot \nabla_v W_{N,t} \left(\frac{x+y}{2}, v \right)\,dv \end{align} $$
$$ \begin{align} &\qquad + N \int e^{i v \cdot \frac{x-y}{\varepsilon}} \left( \nabla ( \eta * \boldsymbol{A}_{\alpha_t})^2 \right) \left( \frac{x + y}{2} \right) \cdot \nabla_v W_{N,t} \left( \frac{x+y}{2}, v \right)\,dv \end{align} $$
$$ \begin{align} &\qquad - 2 \sum_{j=1}^3 N \int e^{i v \cdot \frac{x-y}{\varepsilon}} v^{(j)} \left( \nabla \eta * \boldsymbol{A}_{\alpha_t}^{(j)} \right) \left( \frac{x+y}{2} \right) \cdot \nabla_v W_{N,t} \left( \frac{x+y}{2}, v \right)\,dv. \end{align} $$
By straightforward manipulations the right-hand side can be written in terms of the Weyl quantization of
$W_{N,t}$
. More explicitly, we obtain
$$ \begin{align} (\text{91a}) &= - i \varepsilon^{-1} \left( \left[ - \varepsilon^2 \Delta, \omega_{N,t} \right](x;y) + 2 \eta * \boldsymbol{A}_{\alpha_t} \left( \frac{x + y}{2} \right) \cdot \left[ i \varepsilon \nabla, \omega_{N,t} \right](x;y) \right) , \nonumber \\ (\text{91b}) & = - i \varepsilon^{-1} (\nabla K * \rho_{\omega_t}) \left( \frac{x+y}{2} \right) \cdot (x-y) \, \omega_{N,t}(x;y), \nonumber \\ (\text{91c}) & = - i \varepsilon^{-1} \left( \nabla ( \eta * \boldsymbol{A}_{\alpha_t})^2 \right) \left( \frac{x + y}{2} \right) \cdot (x-y) \, \omega_{N,t}(x;y) , \nonumber \\ (\text{91d}) &= - i \sum_{j=1}^3 \varepsilon^{-1} \left( \nabla \eta * \boldsymbol{A}_{\alpha_t}^{(j)} \right) \left( \frac{x+y}{2} \right) (x - y) \{ i \varepsilon \nabla^{(j)}, \omega_{N,t} \}(x;y). \end{align} $$
Plugging these expressions into the equality from above and multiplying by
$i \varepsilon $
lead to the first equation in (87). Because of
$$ \begin{align} \{ i \varepsilon \nabla , \omega_{N,t} \}(x;y) &= - 2 N \int v W_{N,t} \left( \frac{x+y}{2} , v \right) e^{i v \cdot \frac{x-y}{\varepsilon}}\,dv \end{align} $$
we get
$$ \begin{align} \{ i \varepsilon \nabla , \omega_{N,t} \}(x;x) &= - 2 N \int v W_{N,t} \left( x , v \right)\,dv. \end{align} $$
Together with (90) this enables us to write the charge current in the Vlasov–Maxwell equations (15) as
$$ \begin{align} \widetilde{\boldsymbol{J}}_{W_{N,t}, \alpha_t}(x) &= 2 \int \left( v - (\eta * \boldsymbol{A}_{\alpha_t})(x) \right) W_{N,t}(x,v)\,dv \nonumber \\ &= - N^{-1} \{ i \varepsilon \nabla , \omega_{N,t} \}(x;x) - 2 \rho_{\omega_t}(x) (\eta * \boldsymbol{A}_{\alpha_t})(x) , \end{align} $$
leading to the second equation in (87).
4.2 Grönwall estimate
In this section, we prove Theorem 2.1. This will be obtained by means of a Grönwall type estimate. Throughout this section, let
$(\omega _{N,t}, \alpha _t)$
and
$(\widetilde {\omega }_{N,t}, \widetilde {\alpha }_t)$
denote the solutions of (9) and (87) with initial data
$(\omega _{N,0} , \alpha _0)$
and
$(\widetilde {\omega }_{N,0}, \widetilde {\alpha }_0)$
, respectively.
Using Duhamel’s formula we write the difference between the mode functions of (9) and (87) as
$$ \begin{align} \alpha_t(k,\lambda) - \widetilde{\alpha}_t(k,\lambda) &= e^{- i \left| k \right| t} \left( \alpha_0(k,\lambda) - \widetilde{\alpha}_0(k,\lambda) \right) \nonumber \\ &\quad + i \int_0^t e^{- i \left| k \right| (t-s)} \sqrt{\frac{4 \pi^3}{\left| k \right|}} \mathcal{F}[\eta](k) \boldsymbol{\epsilon}_{\lambda}(k) \cdot \left( \mathcal{F}[\boldsymbol{J}_{\omega_{N,s}, \alpha_s}](k) - \mathcal{F}[\boldsymbol{J}_{\widetilde{\omega}_{N,s}, \widetilde{\alpha}_s}](k) \right)ds. \end{align} $$
By means of (28) and (68) with
$(\omega , \alpha ) = (\widetilde {\omega }_{N,s}, \widetilde {\alpha }_s)$
and
$(\omega ' , \alpha ') = (\omega _{N,s}, \alpha _s)$
we estimate
$$ \begin{align} &\left\| \alpha_t - \widetilde{\alpha}_t \right\| _{\dot{\mathfrak{h}}_{-1/2} \, \cap \, \dot{\mathfrak{h}}_{1/2}} - \left\| \alpha_0 - \widetilde{\alpha}_0 \right\| _{\dot{\mathfrak{h}}_{-1/2} \, \cap \, \dot{\mathfrak{h}}_{1/2}} \nonumber \\ &\quad \leq C \int_0^t \left\| \left( \left| \cdot \right|{}^{-1} + 1 \right) \mathcal{F}[\eta](k) \left( \mathcal{F}[\boldsymbol{J}_{\omega_{N,s}, \alpha_s}] - \mathcal{F}[\boldsymbol{J}_{\widetilde{\omega}_{N,s}, \widetilde{\alpha}_s}] \right) \right\| _{L^2(\mathbb{R}^3)} ds \nonumber \\ &\quad \leq C \int_0^t \left\| \left( \left| \cdot \right|{}^{-1} + 1 \right) \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \left( \mathcal{F}[\boldsymbol{J}_{\omega_{N,s}, \alpha_s}] - \mathcal{F}[\boldsymbol{J}_{\widetilde{\omega}_{N,s}, \widetilde{\alpha}_s}] \right) \right\| _{L^{\infty}(\mathbb{R}^3)}ds \nonumber \\ &\quad \leq C N^{-1} \int_0^t \Big( \big( 1 + \left\| \widetilde{\alpha}_s \right\| _{\dot{\mathfrak{h}}_{1/2}} \big) \left\| \omega_{N,s} - \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} + \left\| \omega_{N,s} \right\| _{\mathfrak{S}^1} \left\| \alpha_s - \widetilde{\alpha}_s \right\| _{\dot{\mathfrak{h}}_{-1/2} \, \cap \, \dot{\mathfrak{h}}_{1/2}} \Big)ds. \end{align} $$
Next, we are going to estimate the difference between particle operators
$\omega _{N,t}$
and
$\widetilde {\omega }_{N,t}$
. To this end, it is convenient to define the operator
$$ \begin{align} \textsf{D} &= \left( 1 - \varepsilon^2 \Delta \right)^{1/2} \end{align} $$
and the
$\varepsilon $
-dependent Sobolev norm
$$ \begin{align} \left\| \omega \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} &= \left\| \left( 1 - \varepsilon^2 \Delta \right)^{1/2} \omega \left( 1 - \varepsilon^2 \Delta \right)^{1/2} \right\| _{\mathfrak{S}^1} = \left\| \textsf{D} \omega \textsf{D} \right\| _{\mathfrak{S}^1}. \end{align} $$
Note that for notational convenience we refrain from indicating the dependence of the operator on
$\varepsilon $
. For a rigorous treatment of the Duhamel formula below we introduce a regularized version of
$\textsf {D}$
which is defined by
$$ \begin{align} \textsf{D}_{\leq n} &= \frac{\left( 1 - \varepsilon^2 \Delta \right)^{1/2}}{\left( 1 - \varepsilon^2 \Delta / n^2 \right)^{1/2}} \quad \text{with} \quad n \in \mathbb{N}. \end{align} $$
Note that
$\textsf {D}_{\leq n}$
is a bounded operator with operator norm
$\left \| \textsf {D}_{\leq n} \right \| _{\mathfrak {S}^{\infty }} \leq n$
which allows us to write the
$\varepsilon $
-dependent Sobolev norm of
$\omega \in \mathfrak {S}^{1,1}$
as
$$ \begin{align} \left\| \omega \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} &= \lim_{n \rightarrow \infty} \left\| \textsf{D}_{\leq n} \omega \textsf{D}_{\leq n} \right\| _{\mathfrak{S}^{1}}. \end{align} $$
In addition, we define the time-dependent Hamiltonian
$$ \begin{align} h(t) = \left( - i \varepsilon \nabla - \eta * \boldsymbol{A}_{\alpha_t} \right)^2 + K * \rho_{\omega_{N,t}} \end{align} $$
and the two parameter group
$U(t;s)$
satisfying
$$ \begin{align} i \varepsilon \partial_t U(t;s) = h(t) U(t;s) \quad \text{and} \quad U(s;s) = 1. \end{align} $$
Moreover, let
$U(t) = U(t;0)$
. Using (9), (87) and the linearity of the mappings
$\alpha \mapsto \boldsymbol {A}_{\alpha }$
and
$\omega \mapsto \rho _{\omega }$
we compute
$$ \begin{align} &i \varepsilon \partial_t \left( U^*(t) \textsf{D}_{\leq n} \left( \omega_{N,t} - \widetilde{\omega}_{N,t} \right) \textsf{D}_{\leq n} U(t) \right) \nonumber \\ &\quad = - U^*(t) \left[ h(t) , \textsf{D}_{\leq n} \left( \omega_{N,t} - \widetilde{\omega}_{N,t} \right) \textsf{D}_{\leq n} \right] U(t) \nonumber \\ &\qquad + U^*(t) \textsf{D}_{\leq n} \left( i \varepsilon \partial_t \left( \omega_{N,t} - \widetilde{\omega}_{N,t} \right) \right) \textsf{D}_{\leq n} U(t) \nonumber \\ &\quad = U^*(t) \textsf{D}_{\leq n} \left[ \left( (\eta * \boldsymbol{A}_{\alpha_t})^2 - (\eta * \boldsymbol{A}_{\widetilde{\alpha}_t})^2 + 2 \eta * \boldsymbol{A}_{\alpha_t - \widetilde{\alpha}_t} \cdot i \varepsilon \nabla \right) , \widetilde{\omega}_{N,t} \right] \textsf{D}_{\leq n} U(t) \nonumber \\ &\qquad + U^*(t) \textsf{D}_{\leq n} \left[ K * \rho_{\omega_{N,t} - \widetilde{\omega}_{N,t}} , \widetilde{\omega}_{N,t} \right] \textsf{D}_{\leq n} U(t) \nonumber \\ &\qquad - U^*(t) \textsf{D}_{\leq n} \left[ X_{\omega_{N,t}} , \omega_{N,t} \right] \textsf{D}_{\leq n} U(t) \nonumber \\ &\qquad + U^*(t) \textsf{D}_{\leq n} \left( \left[ K* \rho_{\widetilde{\omega}_{N,t}}, \widetilde{\omega}_{N,t} \right] - B_t \right) \textsf{D}_{\leq n} U(t) \nonumber \\ &\qquad + U^*(t) \textsf{D}_{\leq n} \left( \left[ (\eta * \boldsymbol{A}_{\widetilde{\alpha}_t})^2 + 2 \eta * \boldsymbol{A}_{\widetilde{\alpha}_t} \cdot i \varepsilon \nabla , \widetilde{\omega}_{N,t} \right] - C_t \right) \textsf{D}_{\leq n} U(t) \nonumber \\ &\qquad - U^*(t) \textsf{D}_{\leq n} \left( \omega_{N,t} - \widetilde{\omega}_{N,t} \right) \left[ h(t) , \textsf{D}_{\leq n} \right] U(t) \nonumber \\ &\qquad - U^*(t) \left[ h(t) , \textsf{D}_{\leq n} \right] \left( \omega_{N,t} - \widetilde{\omega}_{N,t} \right) \textsf{D}_{\leq n} U(t). \end{align} $$
If we then apply Duhamel’s formula, take the trace norm, take the limit
$n \rightarrow \infty $
and use (101), as well as
$\left \| \textsf {D}_{\leq n} \textsf {D}^{-1} \right \| _{\mathfrak {S}^{\infty }} \leq 1$
, we obtain
$$ \begin{align} \left\| \omega_{N,t} - \widetilde{\omega}_{N,t} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} &\leq \left\| \omega_{N,0} - \widetilde{\omega}_{N,0} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \int_0^t \left\| \left[ \left( (\eta * \boldsymbol{A}_{\alpha_s})^2 - (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 \right) , \widetilde{\omega}_{N,s} \right] \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}}ds \end{align} $$
$$ \begin{align} &\quad + 2 \varepsilon^{-1} \int_0^t \left\| \left[ \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \cdot i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}}ds \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \int_0^t \left\| \left[ K * \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}} , \widetilde{\omega}_{N,s} \right] \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}}ds \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \int_0^t \left\| \left[ X_{\omega_{N,s}} , \omega_{N,s} \right] \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}}ds \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \int_0^t \left\| \left[ K* \rho_{\widetilde{\omega}_{N,s}}, \widetilde{\omega}_{N,s} \right] - B_s \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}}ds \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \int_0^t \left\| \left[ (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 + 2 \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \cdot i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] - C_s \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}}ds \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \lim_{n \rightarrow \infty} \int_0^t \left\| \textsf{D} \left( \omega_{N,s} - \widetilde{\omega}_{N,s} \right) \left[ h(s) , \textsf{D}_{\leq n} \right] \right\| _{\mathfrak{S}^1}ds \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \lim_{n \rightarrow \infty} \int_0^t \left\| \left[ h(s) , \textsf{D}_{\leq n} \right] \left( \omega_{N,s} - \widetilde{\omega}_{N,s} \right) \textsf{D} \right\| _{\mathfrak{S}^1}ds. \end{align} $$
In the following, we estimate each line separately.
The term (105b):
Using
$$ \begin{align} \textsf{D} [A, B] \textsf{D} &= [A, \textsf{D} B\textsf{D}] + [\textsf{D},A] B\textsf{D} + \textsf{D} B[\textsf{D}, A] \end{align} $$
and writing the difference of the vector potentials as
$$ \begin{align} (\eta * \boldsymbol{A}_{\alpha_s})^2 - (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 &= \eta * \boldsymbol{A}_{\alpha_s + \widetilde{\alpha}_s} \cdot \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \end{align} $$
we obtain
$$ \begin{align} &\left\| \left[ \left( (\eta * \boldsymbol{A}_{\alpha_s})^2 - (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 \right) , \widetilde{\omega}_{N,s} \right] \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} \nonumber \\ &\quad \leq \left\| \left[ \left( (\eta * \boldsymbol{A}_{\alpha_s})^2 - (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 \right) , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^{1}} \nonumber \\ &\qquad + \left\| \left[ \textsf{D}, \left( (\eta * \boldsymbol{A}_{\alpha_s})^2 - (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 \right) \right] \widetilde{\omega}_{N,s} \textsf{D} \right\| _{\mathfrak{S}^{1}} \nonumber \\ &\qquad + \left\| \textsf{D} \widetilde{\omega}_{N,s} \left[ \textsf{D}, \left( (\eta * \boldsymbol{A}_{\alpha_s})^2 - (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 \right) \right] \right\| _{\mathfrak{S}^{1}} \nonumber \\ &\quad \leq \left\| \eta * \boldsymbol{A}_{\alpha_s + \widetilde{\alpha}_s} \right\| _{\mathfrak{S}^{\infty}} \left\| \left[ \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^{1}} \end{align} $$
$$ \begin{align} &\qquad + \left\| \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \right\| _{\mathfrak{S}^{\infty}} \left\| \left[ \eta * \boldsymbol{A}_{\alpha_s + \widetilde{\alpha}_s} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^{1}} \end{align} $$
$$ \begin{align} &\qquad + \left\| \left[ \textsf{D}, \left( (\eta * \boldsymbol{A}_{\alpha_s})^2 - (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 \right) \right] \right\| _{\mathfrak{S}^{\infty}} \left( \left\| \widetilde{\omega}_{N,s} \textsf{D} \right\| _{\mathfrak{S}^{1}} + \left\| \textsf{D} \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}^{1}} \right). \end{align} $$
Using the Fourier expansion of the regularized vector potential
$$ \begin{align} \eta * \boldsymbol{A}_{\alpha}(x) = 2 \text{Re} \sum_{\lambda = 1,2} \int \frac{\mathcal{F}[\eta](k)}{\sqrt{2 \left| k \right|}} \boldsymbol{\epsilon}_{\lambda}(k) e^{i k x} \alpha(k,\lambda)\,dk, \end{align} $$
(82a), the Cauchy-Schwarz inequality and (28) we estimate
$$ \begin{align} \left\| \left[ \eta * \boldsymbol{A}_{\alpha} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1} &\leq 2 \sum_{\lambda = 1,2} \int \left| k \right|{}^{1/2} \left| \mathcal{F}[\eta](k) \right| \left| \alpha(k,\lambda) \right| \left\| \left[ \hat{x} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1}dk \nonumber \\ &\leq C \left\| \mathcal{F}[\eta](k) \right\| _{L^2(\mathbb{R}^3)} \min \left\{ \left\| \alpha \right\| _{\mathfrak{h}} , \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}} \right\} \left\| \left[ \hat{x} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1} \nonumber \\ &\leq C \min \left\{ \left\| \alpha \right\| _{\mathfrak{h}} , \left\| \alpha \right\| _{\dot{\mathfrak{h}}_{1/2}} \right\} \left\| \left[ \hat{x} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1}. \end{align} $$
Together with (39a) this leads to
$$ \begin{align} (108a) + (108b) &\leq C \left( \left\| \alpha_s \right\| _{\dot{\mathfrak{h}}_{1/2}} + \left\| \widetilde{\alpha}_s \right\| _{\dot{\mathfrak{h}}_{1/2}} \right) \left\| \alpha_s - \widetilde{\alpha}_s \right\| _{\mathfrak{h}} \left\| \left[ \hat{x} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1}. \end{align} $$
By means of (109), (82b), (39a) and (28) we estimate
$$ \begin{align} &\left\| \left[ \textsf{D}, \left( (\eta * \boldsymbol{A}_{\alpha_s})^2 - (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 \right) \right] \right\| _{\mathfrak{S}^{\infty}} \nonumber \\ &\quad \leq \left\| \eta * \boldsymbol{A}_{\alpha_s+ \tilde{\alpha}_s} \right\| _{\mathfrak{S}^{\infty}} \left\| \left[ \textsf{D}, \eta * \boldsymbol{A}_{\alpha_s - \tilde{\alpha}_s} \right] \right\| _{\mathfrak{S}^{\infty}} + \left\| \eta * \boldsymbol{A}_{\alpha_s - \tilde{\alpha}_s} \right\| _{\mathfrak{S}^{\infty}} \left\| \left[ \textsf{D}, \eta * \boldsymbol{A}_{\alpha_s + \tilde{\alpha}_s} \right] \right\| _{\mathfrak{S}^{\infty}} \nonumber \\ &\quad \leq C \varepsilon \left\| \alpha_s + \widetilde{\alpha}_s \right\| _{\mathfrak{h}} \sum_{\lambda = 1,2} \int \left| k \right|{}^{1/2} \left| \mathcal{F}[\eta](k) \right| \left| \alpha_s(k,\lambda) - \widetilde{\alpha}_s(k,\lambda) \right|\,dk \nonumber \\ &\qquad + C \varepsilon \left\| \alpha_s - \widetilde{\alpha}_s \right\| _{\mathfrak{h}} \sum_{\lambda = 1,2} \int \left| k \right|{}^{1/2} \left| \mathcal{F}[\eta](k) \right| \left| \alpha_s(k,\lambda) + \widetilde{\alpha}_s(k,\lambda) \right|\,dk \nonumber \\ &\quad \leq C \varepsilon \left( \left\| \alpha_s \right\| _{\dot{\mathfrak{h}}_{1/2}} + \left\| \widetilde{\alpha}_s \right\| _{\dot{\mathfrak{h}}_{1/2}} \right) \left\| \alpha_s - \widetilde{\alpha}_s \right\| _{\mathfrak{h}}. \end{align} $$
In total, we get
$$ \begin{align} &\left\| \left[ \left( (\eta * \boldsymbol{A}_{\alpha_s})^2 - (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 \right) , \widetilde{\omega}_{N,s} \right] \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} \nonumber \\ &\quad \leq C \left( \left\| \alpha_s \right\| _{\dot{\mathfrak{h}}_{1/2}} + \left\| \widetilde{\alpha}_s \right\| _{\dot{\mathfrak{h}}_{1/2}} \right) \left\| \alpha_s - \widetilde{\alpha}_s \right\| _{\mathfrak{h}} \Big( \left\| \left[ \hat{x} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1} + \varepsilon \left\| \widetilde{\omega}_{N,s} \textsf{D} \right\| _{\mathfrak{S}^{1}} + \varepsilon \left\| \textsf{D} \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}^{1}} \Big). \end{align} $$
Using Lemma B.1 we, finally, get
$$ \begin{align} (105b) &\leq C N \int_0^t \sum_{j=0}^6 \varepsilon^{j} \left\| \widetilde{W}_{N,s} \right\| _{H_6^{j+1}} \left( \left\| \alpha_s \right\| _{\dot{\mathfrak{h}}_{1/2}} + \left\| \widetilde{\alpha}_s \right\| _{\dot{\mathfrak{h}}_{1/2}} \right) \left\| \alpha_s - \widetilde{\alpha}_s \right\| _{\mathfrak{h}}ds. \end{align} $$
The term (105c):
Using (106) and
$\left ( \nabla \cdot \eta * \boldsymbol {A}_{\alpha _s - \widetilde {\alpha }_s} \right ) = 0$
we obtain
$$ \begin{align} \textsf{D} \left[ \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \cdot i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] \textsf{D} &= \left[ \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \cdot i \varepsilon \nabla\right] \nonumber \\ &\quad + \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \cdot \left[ i \varepsilon \nabla , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \nonumber \\ &\quad + \left[ \textsf{D} , \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \right] \cdot i \varepsilon \nabla \widetilde{\omega}_{N,s} \textsf{D} \nonumber \\ &\quad + \textsf{D} \widetilde{\omega}_{N,s} i \varepsilon \nabla \cdot \left[ \textsf{D} , \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \right] \end{align} $$
and
$$ \begin{align} \left\| \left[ \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \cdot i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} &\leq \varepsilon \left\| \left[ \textsf{D} , \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \right] \right\| _{\mathfrak{S}^{\infty}} \left( \left\| \nabla \widetilde{\omega}_{N,s} \textsf{D} \right\| _{\mathfrak{S}^1} + \left\| \textsf{D} \widetilde{\omega}_{N,s} \nabla \right\| _{\mathfrak{S}^1} \right) \nonumber \\ &\quad + \varepsilon \left\| \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \right\| _{\mathfrak{S}^{\infty}} \left\| \left[ \nabla , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1} \nonumber \\ &\quad + \varepsilon \left\| \left[ \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \nabla \right] \right\| _{\mathfrak{S}^1}. \end{align} $$
By (82b), (40a) and (110) with
$\textsf {D} \widetilde {\omega }_{N,s} \textsf {D}$
being replaced by
$\textsf {D}\widetilde {\omega }_{N,s} \textsf {D} \nabla $
we obtain in analogy to the estimates above
$$ \begin{align} \left\| \left[ \eta * \boldsymbol{A}_{\alpha_s - \widetilde{\alpha}_s} \cdot i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} &\leq C \varepsilon \left\| \alpha_s - \widetilde{\alpha}_s \right\| _{\mathfrak{h}} \Big( \left\| \varepsilon \nabla \widetilde{\omega}_{N,s} \textsf{D} \right\| _{\mathfrak{S}^1} + \left\| \textsf{D} \widetilde{\omega}_{N,s} \varepsilon \nabla \right\| _{\mathfrak{S}^1} \nonumber \\ &\qquad \quad + \left\| \left[ \nabla , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1} + \left\| \left[ \hat{x} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \nabla \right] \right\| _{\mathfrak{S}^1} \Big). \end{align} $$
Together with Lemma B.1 this leads to
$$ \begin{align} (105c) &\leq C N \int_0^t \sum_{j=0}^7 \varepsilon^{j} \left\| \widetilde{W}_{N,s} \right\| _{H_7^{j+1}} \left\| \alpha_s - \widetilde{\alpha}_s \right\| _{\mathfrak{h}} ds. \end{align} $$
The term (105d):
Using (106) we obtain
$$ \begin{align} \textsf{D} \left[ K * \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}} , \widetilde{\omega}_{N,s} \right] \textsf{D} &= \textsf{D} \widetilde{\omega}_{N,s} \left[ \textsf{D} , K * \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}} \right] \nonumber \\ &\quad + \left[ \textsf{D} , K * \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}} \right] \widetilde{\omega}_{N,s} \textsf{D} \nonumber \\ &\quad + \left[ K * \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}} , \textsf{D} \, \widetilde{\omega}_{N,s} \textsf{D} \right] \end{align} $$
and estimate
$$ \begin{align} (105d) &\leq \varepsilon^{-1} \int_0^t \left\| \left[ \textsf{D} , K * \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}} \right] \right\| _{\mathfrak{S}^{\infty}} \left( \left\| \textsf{D} \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}^1} + \left\| \widetilde{\omega}_{N,s} \textsf{D} \right\| _{\mathfrak{S}^1} \right)\,ds \nonumber \\ &\quad + \varepsilon^{-1} \int_0^t \left\| \left[ K * \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}} , \textsf{D} \, \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1}ds. \end{align} $$
By means of (82b), (48), (50) and (28) we get
$$ \begin{align} \left\| \left[ \textsf{D} , K * \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}} \right] \right\| _{\mathfrak{S}^{\infty}} &\leq C \varepsilon \int \left| k \right| \left| \mathcal{F}[K](k) \right| \left| \mathcal{F}[\rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}}](k) \right|\,dk \nonumber \\ &\leq C \varepsilon \left\| \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}} \right\| _{L^1(\mathbb{R}^3)} \left\| \left| \cdot \right|{}^{-1/2} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)}^2 \nonumber \\ &\leq C N^{-1} \varepsilon \left\| \omega_{N,s} - \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}^1}. \end{align} $$
Similarly, we get
$$ \begin{align} \left\| \left[ K * \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}} , \textsf{D} \, \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1} &\leq C \left\| \left[ \hat{x} , \textsf{D} \, \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1} \int \left| k \right| \left| \mathcal{F}[K](k) \right| \left| \mathcal{F}[\rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}}](k) \right|\,dk \nonumber \\ &\leq C N^{-1} \varepsilon \left\| \omega_{N,s} - \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}^1} \left\| \left[ \hat{x} , \textsf{D} \, \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1} \end{align} $$
by using (82a) and the Fourier decomposition
$$ \begin{align} \mathcal{F}[K * \rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}}](k) &= (2 \pi)^{3/2} \mathcal{F}[K](k) \mathcal{F}[\rho_{\omega_{N,s} - \widetilde{\omega}_{N,s}}](k). \end{align} $$
Together with Lemma B.1 this shows
$$ \begin{align} (105d) &\leq C N^{-1} \int_0^t \left\| \omega_{N,s} - \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}^1} \left( \left\| \textsf{D} \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}^1} + \left\| \widetilde{\omega}_{N,s} \textsf{D} \right\| _{\mathfrak{S}^1} + \varepsilon^{-1} \left\| \left[ \hat{x} , \textsf{D} \, \widetilde{\omega}_{N,s} \textsf{D} \right] \right\| _{\mathfrak{S}^1} \right)\,ds \nonumber \\ &\leq C \int_0^t \left\| \omega_{N,s} - \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}^1} \sum_{j=0}^6 \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_6^{j+1}}ds. \end{align} $$
The term (105e):
Using (53) we estimate
$$ \begin{align} (105e) &= \varepsilon^{-1} \int_0^t \left\| \textsf{D} \left[ X_{\omega_{N,s}} , \omega_{N,s} \right] \textsf{D} \right\| _{\mathfrak{S}^{1}}ds \nonumber \\ &\leq \varepsilon^{-1} N^{-1} \int_0^t \int \left| \mathcal{F}[K](k) \right| \left\| \textsf{D} \left[ e^{i k \hat{x}} \omega_{N,s} e^{- i k \hat{x}} , \omega_{N,s} \right] \textsf{D} \right\| _{\mathfrak{S}^{1}} dk\,ds \nonumber \\ &\leq \varepsilon^{-1} N^{-1} \int_0^t \int \left| \mathcal{F}[K](k) \right| \bigg( \left\| \textsf{D} e^{i k \hat{x}} \omega_{N,s} e^{- i k \hat{x}} \omega_{N,s} \textsf{D} \right\| _{\mathfrak{S}^{1}} \nonumber \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad + \left\| \textsf{D} \omega_{N,s} e^{i k \hat{x}} \omega_{N,s} e^{- i k \hat{x}} \textsf{D} \right\| _{\mathfrak{S}^{1}} \bigg)\,dk\,ds \nonumber \\ &\leq \varepsilon^{-1} N^{-1} \int_0^t \left\| \omega_{N,s}^{1/2} \right\| _{\mathfrak{S}^{\infty}}^2 \int \left| \mathcal{F}[K](k) \right| \bigg( \left\| \textsf{D} e^{i k \hat{x}} \omega_{N,s}^{1/2} \right\| _{\mathfrak{S}^{2}} \left\| \omega_{N,s}^{1/2} \textsf{D} \right\| _{\mathfrak{S}^{2}} \nonumber \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad + \left\| \textsf{D} \omega_{N,s}^{1/2} \right\| _{\mathfrak{S}^{2}} \left\| \omega_{N,s}^{1/2} e^{i k \hat{x}} \textsf{D} \right\| _{\mathfrak{S}^{2}} \bigg)\,dk\,ds. \end{align} $$
By means of
$\left \| \textsf {D} \omega _{N,s}^{1/2} \right \| _{\mathfrak {S}^{2}}^2 = \text {Tr} \left ( \textsf {D}^2 \omega _{N,s} \right )$
and
$$ \begin{align} \left\| \textsf{D} e^{i k \hat{x}} \omega_{N,s}^{1/2} \right\| _{\mathfrak{S}^{2}}^2 &= \text{Tr} \left( \omega_{N,s} e^{- i k \hat{x} } \textsf{D}^2 e^{i k \hat{x}} \right) \nonumber \\ &= \text{Tr} \left( \left( 1 - \varepsilon^2 \Delta + \varepsilon^2 k^2 - 2 \varepsilon k \cdot i \varepsilon \nabla \right) \omega_{N,s} \right) \nonumber \\ &\leq 2 \text{Tr} \left( \left( 1 - \varepsilon^2 \Delta + \varepsilon^2 k^2 \right) \omega_{N,s} \right) \nonumber \\ &= 2 \text{Tr} \left( \left( \textsf{D}^2 + \varepsilon^2 k^2 \right) \omega_{N,s} \right), \end{align} $$
together with (48) and (28) we obtain
$$ \begin{align} (105e) &\leq C \varepsilon^{-1} N^{-1} \int_0^t \left\| \omega_{N,s}^{1/2} \right\| _{\mathfrak{S}^{\infty}}^2 \int \left| \mathcal{F}[K](k) \right| \sqrt{\text{Tr} \left( \left( \textsf{D}^2 + \varepsilon^2 k^2 \right) \omega_{N,s} \right)} \sqrt{\text{Tr} \left( \textsf{D}^2 \omega_{N,s} \right)}\,dk\,ds \nonumber \\ &\leq C \varepsilon^{-1} N^{-1} \int_0^t \left\| \omega_{N,s}^{1/2} \right\| _{\mathfrak{S}^{\infty}}^2 \Big( \text{Tr} \left( \omega_{N,s} \right) + \text{Tr} \left( - \varepsilon^2 \Delta \omega_{N,s} \right) \Big)\,ds. \end{align} $$
Note that
$\omega _{N,s}$
is a trace class operator, ensuring the existence of a spectral set
$\{ \lambda _j, \varphi _j \}_{j \in \mathbb {N}}$
with
$\lambda _j \geq 0$
for all
$j \in \mathbb {N}$
such that
$\omega _{N,s} = \sum _{j \in \mathbb {N}} | \varphi _j \rangle \langle \varphi _j |$
. This allows us to estimate the operator norm of
$\omega _{N,s}$
by
$$ \begin{align} \left\| \omega_{N,s}^{1/2} \right\| _{\mathfrak{S}^{\infty}}^2 &\leq \sup_{j \in \mathbb{N}} \{ \lambda_j \} \leq \Big( \sum_{j \in \mathbb{N}} \lambda_j^3 \Big)^{1/3} = \left\| \omega_{N,s} \right\| _{\mathfrak{S}^3} \end{align} $$
and leads to
$$ \begin{align} (105e) &\leq C \varepsilon^{-1} N^{-1} \int_0^t \left\| \omega_{N,s} \right\| _{\mathfrak{S}^3} \Big( \left\| \omega_{N,s} \right\| _{\mathfrak{S}^1} + \text{Tr} \left( - \varepsilon^2 \Delta \omega_{N,s} \right) \Big)\,ds. \end{align} $$
With this regard note that
$\left \| \omega _{N,s} \right \| _{\mathfrak {S}^3} = \left \| \omega _{N,0} \right \| _{\mathfrak {S}^3}$
because of
$$ \begin{align} i \varepsilon \frac{d}{dt} \left\| \omega_{N,s} \right\| _{\mathfrak{S}^3}^3 &= \text{Tr} \left( \left[ \left( - i \varepsilon \nabla - \eta * \boldsymbol{A}_{\alpha_s} \right)^2 + K * \rho_{\omega_s} - X_{\omega_s} , \omega_{N,s}^3 \right] \right) = 0 \end{align} $$
and that
$\left \| \omega _{N,0} \right \| _{\mathfrak {S}^3} = \text {Tr} \left ( \omega _{N,0}^3 \right )^{1/3} \leq \left \| \omega _{N,0} \right \| _{\mathfrak {S}^{\infty }}^{2/3} \left \| \omega _{N,0} \right \| _{\mathfrak {S}^{1}}^{1/3}$
. In combination with (69b), the conservation of mass and energy, and the assumptions on the initial data we get
$$ \begin{align} (105e) &\leq C \varepsilon^{-1} N^{-2/3} \left( \mathcal{E}^{\mathrm{{MS}}}[\omega_0,\alpha_0] + N \right) t. \end{align} $$
The term (105f):
The next term will be estimated by similar means as in [Reference Benedikter, Porta, Saffirio and Schlein11, Chapter 3]. Using the fact that the Hilbert-Schmidt norm of an operator is related to the
$L^2$
norm of its integral kernel, we obtain
$$ \begin{align} \left\| \left( 1 - \varepsilon^2 \Delta \right)^{-1} \left( 1 + x^2 \right)^{-1} \right\| _{\mathfrak{S}^2} &\leq C \sqrt{N}. \end{align} $$
Together with Hölder’s inequality for Schatten spaces, this leads to
$$ \begin{align} (105f) &\leq C N^{1/2} \varepsilon^{-1} \int_0^t \left\| \left( 1 + x^2 \right) \textsf{D}^3 \left( \left[ K* \rho_{\widetilde{\omega}_{N,s}}, \widetilde{\omega}_{N,s} \right] - B_s \right) \textsf{D} \right\| _{\mathfrak{S}^{2}}ds. \end{align} $$
Since
$$ \begin{align} \left\| A\textsf{D} \right\| _{\mathfrak{S}^2}^2 &= \left\| A \right\| _{\mathfrak{S}^2}^2 + \left\| A i \varepsilon \nabla \right\| _{\mathfrak{S}^2}^2 , \end{align} $$
$$ \begin{align} \left\| \textsf{D} A \right\| _{\mathfrak{S}^2}^2 &= \left\| A \right\| _{\mathfrak{S}^2}^2 + \left\| i \varepsilon \nabla A \right\| _{\mathfrak{S}^2}^2 , \end{align} $$
$$ \begin{align} \left\| \left[\textsf{D} , (1+ x^2) \right] f \right\| _{L^2(\mathbb{R}^3)} &\leq C \left\| \left( 1 + x^2 \right) f \right\| _{L^2(\mathbb{R}^3)} \end{align} $$
and
$$ \begin{align} \left\| i \varepsilon \nabla \left( 1 + x^2 \right) f \right\| _{L^2(\mathbb{R}^3)} &\leq \left\| \left( 1 + x^2 \right) f \right\| _{L^2(\mathbb{R}^3)} + \left\| \left( 1 + x^2 \right) i \varepsilon \nabla f \right\| _{L^2(\mathbb{R}^3)} \end{align} $$
we have
$$ \begin{align} &\left\| \left( 1 + x^2 \right) \textsf{D}^3 \left( \left[ K* \rho_{\widetilde{\omega}_{N,s}}, \widetilde{\omega}_{N,s} \right] - B_s \right) \textsf{D} \right\| _{\mathfrak{S}^{2}} \nonumber \\ &\quad \leq C \left\| \left( 1 + x^2 \right) \textsf{D}^2 \left( \left[ K* \rho_{\widetilde{\omega}_{N,s}}, \widetilde{\omega}_{N,s} \right] - B_s \right) \right\| _{\mathfrak{S}^{2}} \nonumber \\ &\qquad + C \left\| \left( 1 + x^2 \right) i \varepsilon \nabla \textsf{D}^2 \left( \left[ K* \rho_{\widetilde{\omega}_{N,s}}, \widetilde{\omega}_{N,s} \right] - B_s \right) \right\| _{\mathfrak{S}^{2}} \nonumber \\ &\qquad + C \left\| \left( 1 + x^2 \right) \textsf{D}^2 \left( \left[ K* \rho_{\widetilde{\omega}_{N,s}}, \widetilde{\omega}_{N,s} \right] - B_s \right) i \varepsilon \nabla \right\| _{\mathfrak{S}^{2}} \nonumber \\ &\qquad + C \left\| \left( 1 + x^2 \right) i \varepsilon \nabla \textsf{D}^2 \left( \left[ K* \rho_{\widetilde{\omega}_{N,s}}, \widetilde{\omega}_{N,s} \right] - B_s \right) i \varepsilon \nabla \right\| _{\mathfrak{S}^{2}}. \end{align} $$
If we use a second-order Taylor expansion, as in [Reference Benedikter, Porta, Saffirio and Schlein11, Chapter 3], we obtain
$$ \begin{align} &K * \rho_{\widetilde{\omega}_{N,s}}(x) - K * \rho_{\widetilde{\omega}_{N,s}}(y) - \nabla (K * \widetilde{\rho}_s) \left( \frac{x +y}{2} \right) \cdot (x-y) \nonumber \\ &\quad = \sum_{i,j= 1}^3 \int_0^1 \int_0^1 \left( \partial_i \partial_j K * \rho_{\widetilde{\omega}_{N,s}} \right) \big( v_{\lambda,\mu}(x,y) \big) (x-y)_i (x-y)_j \left( \lambda - 1/2 \right)\,d\mu\,d\lambda , \end{align} $$
where
$v_{\lambda ,\mu }(x,y) = \mu (\lambda x + (1 - \lambda ) y) + (1 - \mu ) (x+y)/2$
, we get
$$ \begin{align} &\left( \left[ K* \rho_{\widetilde{\omega}_{N,s}}, \widetilde{\omega}_{N,s} \right] - B_s \right)(x; y) \nonumber \\ &\quad = \sum_{i,j= 1}^3 \int_0^1 \int_0^1 \left( \lambda - 1/2 \right) \left( \partial_i \partial_j K * \rho_{\widetilde{\omega}_{N,s}} \right) \big( v_{\lambda,\mu}(x,y) \big) \left[ \hat{x}_i , \left[ \hat{x}_j , \widetilde{\omega}_{N,s} \right] \right](x;y)\,d\mu\,d\lambda. \end{align} $$
Using
$\left \| O \right \| _{\mathfrak {S}^2}^2 = \iint \left | O(x;y) \right |{}^2 dx\,dy$
for any Hilbert-Schmidt operator O, the product rule for the gradient and estimating the term involving the potential by the
$L^{\infty }(\mathbb {R}^3)$
-norm we get
$$ \begin{align} (105f) &\leq C N^{1/2} \varepsilon^{-1} \int_0^t \sum_{2 \leq \left| \beta \right| \leq 6} \varepsilon^{\left| \beta \right| - 2} \left\| \nabla^{\beta} K * \rho_{\widetilde{\omega}_{N,s}} \right\| _{L^{\infty}(\mathbb{R}^3)} \nonumber \\ &\qquad \times \sum_{\left| \gamma \right| \leq 3} \varepsilon^{\left| \gamma \right|} \left( \left\| \left( 1 + x^2 \right) \nabla^{\gamma} \left[ \hat{x} , \left[ \hat{x} , \widetilde{\omega}_{N,s} \right] \right] \right\| _{\mathfrak{S}^{2}} + \left\| \left( 1 + x^2 \right) \nabla^{\gamma} \left[ \hat{x} , \left[ \hat{x} , \widetilde{\omega}_{N,s} \right] \right] i \varepsilon \nabla \right\| _{\mathfrak{S}^{2}} \right)\,ds. \end{align} $$
Together with (90), (46b) Lemma B.1 this leads to
$$ \begin{align} (105f) &\leq C N \varepsilon^{-1} \int_0^t \sum_{2 \leq \left| \beta \right| \leq 6} \varepsilon^{\left| \beta \right| - 2} \left\| \nabla^{\beta} K * \widetilde{\rho}_{\widetilde{W}_{N,s}} \right\| _{L^{\infty}(\mathbb{R}^3)} \sum_{j=2}^{8} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{6}^{j}}ds \nonumber \\ &\leq C N \varepsilon \int_0^t \left( \left< \varepsilon \right> \left\| \widetilde{W}_{N,s} \right\| _{L^1(\mathbb{R}^6)} + \sum_{i=0}^3 \varepsilon^{2 + i} \left\| \widetilde{W}_{N,s} \right\| _{H_{4}^{j+1}} \right) \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{6}^{j+2}}ds. \end{align} $$
The term (105g):
In the following it will be convenient to split the operator
$C_s$
, defined in (89), into the following three parts
$$ \begin{align} C_{1,s}(x;y) &= \left(\nabla (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 \right) \left(\frac{x+y}{2} \right) \cdot (x - y) \, \widetilde{\omega}_{N,s}(x;y) , \nonumber \\ C_{2,s}(x;y) &= \sum_{j=1}^3 \left( \nabla^{(j)} \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \right) \left( \frac{x + y}{2} \right) \cdot (x - y)^{(j)} \left\{ i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right\}(x;y) , \nonumber \\ C_{3,s}(x;y) &= 2 \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \left( \frac{x + y}{2} \right) \cdot \left[ i \varepsilon \nabla, \widetilde{\omega}_{N,s} \right](x;y). \end{align} $$
Moreover, note that
$$ \begin{align} \left[ 2 \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \cdot i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] &= \left[ \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} , \left\{ i \varepsilon \nabla, \widetilde{\omega}_{N,s} \right\} \right] + \left\{ \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} , \left[ i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] \right\} \end{align} $$
leading to
$$ \begin{align} (105g) &\leq \varepsilon^{-1} \int_0^t \left\| \left[ (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 , \widetilde{\omega}_{N,s} \right] - C_{1,s} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} ds \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \int_0^t \left\| \left[ \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} , \left\{ i \varepsilon \nabla, \widetilde{\omega}_{N,s} \right\} \right] - C_{2,s} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} ds \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \int_0^t \left\| \left\{ \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} , \left[ i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] \right\} - C_{3,s} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} ds. \end{align} $$
The first term can be estimated in complete analogy to (105f). Replacing
$K * \rho _{\widetilde {\omega }_{N,s}}$
by
$(\eta * \boldsymbol {A}_{\widetilde {\alpha }_s})^2$
in (105f), performing exactly the same estimates and applying (40b) lead to
$$ \begin{align} (145a) &\leq C N \varepsilon \int_0^t \sum_{2 \leq \left| \beta \right| \leq 6} \varepsilon^{\left| \beta \right| - 2} \left\| \nabla^{\beta} (\eta * \boldsymbol{A}_{\widetilde{\alpha}_s})^2 \right\| _{L^{\infty}(\mathbb{R}^3)} \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{6}^{j+2}}ds \nonumber \\ &\leq C N \varepsilon \int_0^t \sum_{i=0}^4 \varepsilon^i \left\| \widetilde{\alpha}_s \right\| _{\mathfrak{h}_{i+1}}^2 \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{6}^{j+2}}ds. \end{align} $$
If we replace
$K * \rho _{\widetilde {\omega }_{N,s}}$
by
$\eta * \boldsymbol {A}_{\widetilde {\alpha }_s}$
in (139) we get
$$ \begin{align} &\left( \left[ \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} , \left\{ i \varepsilon \nabla, \widetilde{\omega}_{N,s} \right\} \right] - C_{2,s} \right)(x; y) \nonumber \\ &\quad = \sum_{i,j= 1}^3 \int_0^1 \int_0^1 \left( \lambda - \frac{1}{2} \right) \left( \partial_i \partial_j \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \right) \big( v_{\lambda,\mu}(x,y) \big) \left[ \hat{x}_i , \left[ \hat{x}_j , \left\{ i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right\} \right] \right](x;y)\,d\mu\,d\lambda, \end{align} $$
with
$v_{\lambda ,\mu }(x,y) = \mu (\lambda x + (1 - \lambda ) y) + (1 - \mu ) (x+y)/2$
. By the same estimates as for (105f), Lemma B.1 and (40b) we conclude
$$ \begin{align} (145b) &\leq C N^{1/2} \varepsilon^{-1} \int_0^t \sum_{2 \leq \left| \beta \right| \leq 6} \varepsilon^{\left| \beta \right| - 2} \left\| \nabla^{\beta} \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \right\| _{L^{\infty}(\mathbb{R}^3)} \nonumber \\ &\quad \times\, \sum_{\left| \gamma \right| \leq 3} \varepsilon^{\left| \gamma \right|} \bigg( \left\| \left( 1 + x^2 \right) \nabla^{\gamma} \left[ \hat{x} , \left[ \hat{x} , \left\{ i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right\} \right] \right] \right\| _{\mathfrak{S}^{2}} \nonumber \\ &\quad +\, \left\| \left( 1 + x^2 \right) \nabla^{\gamma} \left[ \hat{x} , \left[ \hat{x} , \left\{ i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right\} \right] \right] i \varepsilon \nabla \right\| _{\mathfrak{S}^{2}} \bigg) ds \nonumber \\ &\leq C N \varepsilon \int_0^t \sum_{2 \leq \left| \beta \right| \leq 6} \varepsilon^{\left| \beta \right| - 2} \left\| \nabla^{\beta} \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \right\| _{L^{\infty}(\mathbb{R}^3)} \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{7}^{j+2}} ds \nonumber \\ &\leq C N \varepsilon \int_0^t \sum_{i=0}^4 \varepsilon^i \left\| \widetilde{\alpha}_s \right\| _{\mathfrak{h}_{i+1}} \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{7}^{j+2}}ds. \end{align} $$
Now let
$u_{\lambda }(x;y) = \lambda x + (1 - \lambda ) (x+y)/2$
. Then,
$$ \begin{align} \eta * \boldsymbol{A}_{\widetilde{\alpha}_s}(x) - \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \left( \frac{x+y}{2} \right) &= \frac{1}{2} \sum_{j=1}^3 (x-y)_j \int_0^1 \left( \partial_j \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \right) \big( u_{\lambda}(x;y) \big)d\lambda \end{align} $$
and
$$ \begin{align} &\left( \left\{ \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} , \left[ i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] \right\} - C_{3,s} \right)(x;y) \nonumber \\ &\quad = \frac{1}{2} \sum_{j=1}^3 \int_0^1 \left( \left( \partial_j \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \right) \big( u_{\lambda}(x;y) \big) - \left( \partial_j \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \right) \big(u_{\lambda}(y;x) \big) \right) \left[ \hat{x}_j , \left[ i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] \right](x;y)\,d\lambda. \end{align} $$
By the same arguments as before we get
$$ \begin{align} (145c) &\leq C N^{1/2} \varepsilon^{-1} \int_0^t \sum_{1 \leq \left| \beta \right| \leq 5} \varepsilon^{\left| \beta \right| - 1} \left\| \nabla^{\beta} \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \right\| _{L^{\infty}(\mathbb{R}^3)} \nonumber \\ & \quad\times\, \sum_{\left| \gamma \right| \leq 3} \varepsilon^{\left| \gamma \right|} \bigg( \left\| \left( 1 + x^2 \right) \nabla^{\gamma} \left[ \hat{x} , \left[ i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] \right] \right\| _{\mathfrak{S}^{2}} \nonumber \\ &\quad +\, \left\| \left( 1 + x^2 \right) \nabla^{\gamma} \left[ \hat{x} , \left[ i \varepsilon \nabla , \widetilde{\omega}_{N,s} \right] \right] i \varepsilon \nabla \right\| _{\mathfrak{S}^{2}} \bigg) ds \nonumber \\ &\leq C N \varepsilon \int_0^t \sum_{1 \leq \left| \beta \right| \leq 5} \varepsilon^{\left| \beta \right| - 1} \left\| \nabla^{\beta} \eta * \boldsymbol{A}_{\widetilde{\alpha}_s} \right\| _{L^{\infty}(\mathbb{R}^3)} \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{6}^{j+2}} ds \nonumber \\ &\leq C N \varepsilon \int_0^t \sum_{i=0}^4 \varepsilon^i \left\| \widetilde{\alpha}_s \right\| _{\mathfrak{h}_i} \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{6}^{j+2}}ds. \end{align} $$
In total this shows
$$ \begin{align} (105g) &\leq C N \varepsilon \int_0^t \sum_{i=0}^4 \varepsilon^i \left\| \widetilde{\alpha}_s \right\| _{\mathfrak{h}_{i+1}} \left( 1 + \left\| \widetilde{\alpha}_s \right\| _{\mathfrak{h}_{i+1}} \right) \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{7}^{j+2}}ds. \end{align} $$
The terms (105h) and (105i):
Using
$$ \begin{align} \left[ \textsf{D}_{\leq n}, h(s) \right] &= 2 i \varepsilon \nabla \cdot \left[ \textsf{D}_{\leq n} , \eta * \boldsymbol{A}_{\alpha_s} \right] + \left[ \textsf{D}_{\leq n} , ( \eta * \boldsymbol{A}_{\alpha_s})^2 \right] + \left[ \textsf{D}_{\leq n}, K * \rho_{\omega_{N,s}} \right] , \end{align} $$
$\left \| \textsf {D}^{-1} i \varepsilon \nabla \right \| _{\mathfrak {S}^{\infty }} = \left \| \left ( 1 - \varepsilon ^2 \Delta \right )^{-1/2} i \varepsilon \nabla \right \| _{\mathfrak {S}^{\infty }} \leq 1$
and
$\left \| \textsf {D}^{-1} \right \| _{\mathfrak {S}^{\infty }} \leq 1$
we get the bound
$$ \begin{align} (105h)+ (105i) &\leq C \varepsilon^{-1} \lim_{n \rightarrow \infty} \int_0^t \left\| \omega_{N,s} - \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} \Big( \left\| \left[ \textsf{D}_{\leq n} , \eta * \boldsymbol{A}_{\alpha_s} \right] \right\| _{\mathfrak{S}^{\infty}} \nonumber \\ &\qquad \qquad + \left\| \left[ \textsf{D}_{\leq n} , (\eta * \boldsymbol{A}_{\alpha_s})^2 \right] \right\| _{\mathfrak{S}^{\infty}} + \left\| \left[ \textsf{D}_{\leq n} , K * \rho_{\omega_{N,s}} \right] \right\| _{\mathfrak{S}^{\infty}} \Big)\,ds. \end{align} $$
Thus if we use (82c),
$\mathcal {F}[K * \rho _{\omega _{N,s}}](k)= \mathcal {F}[K](k) \mathcal {F}[\rho _{\omega _{N,s} }](k)$
and the Fourier expansion of the regularized vector potential (109) we obtain
$$ \begin{align} (105h)+ (105i) &\leq C \int_0^t \left\| \omega_{N,s} - \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} \nonumber \\ &\qquad \times \bigg[ \left( 1 + \left\| \eta * \boldsymbol{A}_{\alpha_s} \right\| _{L^{\infty}(\mathbb{R}^3)} \right) \sum_{\lambda=1,2} \int \left| k \right|{}^{1/2} \left| \mathcal{F}[\eta](k) \right| \left| \alpha_s(k,\lambda) \right| dk \nonumber \\ &\qquad \quad +\int \left| k \right| \left| \mathcal{F}[K](k) \right| \left| \mathcal{F}[\rho_{\omega_{N,s} }](k) \right| dk \bigg]\,ds. \end{align} $$
By means of (39a), (48), (50), and (28) the Cauchy–Schwarz inequality the right-hand side can be bounded by
$$ \begin{align} (105h)+ (105i) &\leq C \int_0^t \left\| \omega_{N,s} - \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} \Big[ \left( 1 + \left\| \alpha_s \right\| _{\dot{\mathfrak{h}}_{1/2}} \right) \left\| \alpha_s \right\| _{\dot{\mathfrak{h}}_{1/2}} + N^{-1} \left\| \omega_{N,s} \right\| _{\mathfrak{S}^1} \Big]ds. \end{align} $$
Collecting the estimates and combining them with (97) lead to
$$ \begin{align} &N^{-1} \left\| \omega_{N,t} - \widetilde{\omega}_{N,t} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} + \left\| \alpha_t - \widetilde{\alpha}_t \right\| _{\dot{\mathfrak{h}}_{-1/2} \, \cap \, \dot{\mathfrak{h}}_{1/2}} \nonumber \\ &\quad\leq C \int_0^t C(s) \left( N^{-1} \left\| \omega_{N,s} - \widetilde{\omega}_{N,s} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} + \left\| \alpha_s - \widetilde{\alpha}_s \right\| _{\dot{\mathfrak{h}}_{-1/2} \, \cap \, \dot{\mathfrak{h}}_{1/2}} \right)ds \nonumber \\ &\qquad + N^{-1} \left\| \omega_{N,0} - \widetilde{\omega}_{N,0} \right\| _{\mathfrak{S}_{\varepsilon}^{1,1}} + \left\| \alpha_0 - \widetilde{\alpha}_0 \right\| _{\dot{\mathfrak{h}}_{-1/2} \, \cap \, \dot{\mathfrak{h}}_{1/2}} + C \varepsilon \int_0^t \widetilde{C}(s)ds , \end{align} $$
with
$$ \begin{align} C(s) &= \bigg( 1 + N^{-1} \left\| \omega_{N,s} \right\| _{\mathfrak{S}^1} + \sum_{j=0}^{7} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{7}^{j+1}} \bigg) \left( 1 + \left\| \alpha_s \right\| _{\dot{\mathfrak{h}}_{1/2}} +\left\| \widetilde{\alpha}_s \right\| _{\dot{\mathfrak{h}}_{1/2}} \right)^2 \end{align} $$
and
$$ \begin{align} \widetilde{C}(s) &= N^{-2/3} \varepsilon^{-2} \left( 1 + N^{-1} \mathcal{E}^{\mathrm{{MS}}}[\omega_0, \alpha_0] \right) + \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{7}^{j+2}} \bigg( \left< \varepsilon \right> \left\| \widetilde{W}_{N,s} \right\| _{L^1(\mathbb{R}^6)} \nonumber \\ &\quad + \sum_{k=0}^3 \varepsilon^{2 + k} \left\| \widetilde{W}_{N,s} \right\| _{H_{4}^{k+1}} + \sum_{k=0}^4 \varepsilon^k \left( 1 + \left\| \widetilde{\alpha}_s \right\| _{\mathfrak{h}_{k+1}} \right)^2 \bigg). \end{align} $$
If we use
$\varepsilon = N^{-1/3}, \left \| \omega _{N,0} \right \| _{\mathfrak {S}^1} = N$
, (69a), (75a) and the conservation of mass and energy both of the Maxwell–Schrödinger and the Vlasov–Maxwell system, we can bound the above quantities by
$$ \begin{align} C(s) &\leq C \bigg( 1 + \sum_{j=0}^{7} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{7}^{j+1}} \bigg) \left( 1 + N^{-1} \mathcal{E}^{\mathrm{{MS}}}[\omega_{N,0}, \alpha_0] + \mathcal{E}^{\mathrm{{VM}}}[\widetilde{W}_{N,0}, \widetilde{\alpha}_0] + \left\| \widetilde{W}_{N,0} \right\| _{L^1(\mathbb{R}^6)} \right)^2 \end{align} $$
and
$$ \begin{align} \widetilde{C}(s) &\leq 1 + N^{-1} \mathcal{E}^{\mathrm{{MS}}}[\omega_0, \alpha_0] + \sum_{j=0}^{6} \varepsilon^j \left\| \widetilde{W}_{N,s} \right\| _{H_{7}^{j+2}} \bigg( \left< \varepsilon \right> \left\| \widetilde{W}_{N,0} \right\| _{L^1(\mathbb{R}^6)} \nonumber \\ &\quad + \sum_{k=0}^3 \varepsilon^{2 + k} \left\| \widetilde{W}_{N,s} \right\| _{H_{4}^{k+1}} + \sum_{k=0}^4 \varepsilon^k \left( 1 + \left\| \widetilde{\alpha}_s \right\| _{\mathfrak{h}_{k+1}} \right)^2 \bigg). \end{align} $$
5 Well-posedness of the Maxwell–Schrödinger equations
In this section, we prove Proposition 2.1. To this end, we recall propagation estimates for the time evolution of the Schrödinger equation with a magnetic Laplacian from [Reference Nakamura and Wada42]. We then consider a linearized version of (9) and show the existence as well as regularity properties of the respective solutions. These will be used to prove the existence of local solutions to (9). Finally, we provide propagation estimates allowing us to conclude that under suitable initial conditions the solutions of the Maxwell–Schrödinger equations exist globally. The strategy of the proof is inspired by [Reference Nakamura and Wada42] and [Reference Petersen44, Reference Antonelli, D’Amico and Marcati3, Reference Antonelli, Marcati and Scandone4].
Lemma 5.1. Let
$\alpha \in L^{\infty }(0,T; \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2}) \cap W^{1,1}(0,T; \dot {\mathfrak {h}}_{-1/2} )$
. The equation
$$ \begin{align} \begin{cases} i \varepsilon \partial_t \psi(t)= \left( - i \varepsilon \nabla - \eta * \boldsymbol{A}_{\alpha} \right)^2 \psi(t) \\ \psi(0) = \psi_0 \end{cases} \end{align} $$
has a unique
$C(0,T; H^2(\mathbb {R}^3)) \cap C^1(0,T; L^2(\mathbb {R}^3))$
solution. There exists a constant
$K_{\varepsilon }$
(depending on
$\varepsilon $
) and a strongly continuous two-parameter family
$U_{\alpha }$
of operators on
$H^s(\mathbb {R}^3)$
,
$0 \leq s \leq 2$
, with
$$ \begin{align} K_{\varepsilon, \alpha,T} &= \sup_{t,\tau \in [0,T]} \left\| U_{\alpha}(t,\tau) \right\| _{\mathfrak{S}^{\infty}(H^2(\mathbb{R}^3))} \leq K_{\varepsilon} \left( 1 + \left\| \alpha \right\| _{L_T^{\infty} \mathfrak{h}_{1/2} \cap L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}}^4 \right) e^{K_{\varepsilon} \left\| \partial_t \alpha \right\| _{L_T^1 \dot{\mathfrak{h}}_{-1/2}}} \end{align} $$
and
$\sup _{t,\tau \in [0,T]} \left \| U_{\alpha }(t,\tau ) \right \| _{\mathfrak {S}^{\infty }(H^s(\mathbb {R}^3))} \leq K_{\varepsilon , \alpha , T}^{s/2}$
. In particular,
$U_\alpha (t,\tau )$
is a unitary group on
$L^2(\mathbb {R}^3)$
and
$U_\alpha ^*(t, \tau ) = U_\alpha (\tau , t)$
. Moreover, for any
$\psi _0 \in H^s(\mathbb {R}^3)$
,
$U_\alpha (t,t_0) \psi _0$
is a unique
$H^s$
-solution to (162).
Proof. Using (39b) we estimate
$$ \begin{align} \left\| \eta * \boldsymbol{A}_{\alpha} \right\| _{L_T^{\infty} H^1(\mathbb{R}^3)} &= \left\| \eta * \boldsymbol{A}_{\left< \cdot \right> \alpha} \right\| _{L_T^{\infty} L^2(\mathbb{R}^3)} \leq \left\| \alpha \right\| _{L_T^{\infty} \mathfrak{h}_{1/2} \cap L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}}. \end{align} $$
By interpolation we get
$\left \| \eta * \boldsymbol {A}_{\alpha _t} \right \| _{L^3(\mathbb {R}^3)} \leq \left \| \alpha _t \right \| _{\dot {\mathfrak {h}}_{-1/2}}$
from (39a) and (39b), leading to
$$ \begin{align} \left\| \eta * \boldsymbol{A}_{\alpha} \right\| _{L^1_T L^3(\mathbb{R}^3)} \leq C \left\| \alpha \right\| _{L_T^1 \dot{\mathfrak{h}}_{-1/2}} \quad \text{and} \quad \left\| \partial_t \eta * \boldsymbol{A}_{\alpha} \right\| _{L^1_T L^3(\mathbb{R}^3)} \leq C \left\| \partial_t \alpha \right\| _{L_T^1 \dot{\mathfrak{h}}_{-1/2}}. \end{align} $$
Since the vector potential is divergence free by construction, this allows us to conclude that Assumption (A1) on [Reference Nakamura and Wada42, p. 574] is satisfied for
$\eta * \boldsymbol {A}_{\alpha }$
and
$u =0$
. Application of [Reference Nakamura and Wada42, Lemma 3.1 and Lemma 3.2] then shows the claim.
5.1 Linear equations
Lemma 5.2. Let
$T>0$
,
$(\omega , \alpha ) \in \left ( C \left ( 0,T; \mathfrak {S}^{1} \left ( L^2(\mathbb {R}^3) \right ) \right ) \cap L^{\infty }(0,T; \mathfrak {S}^{2,1}(L^2(\mathbb {R}^3)) \big ) \right ) \times \left ( L^{\infty }(0,T; \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2}) \cap W^{1,1}(0,T; \dot {\mathfrak {h}}_{-1/2} ) \right )$
and
$\Gamma _0 \in \mathfrak {S}^{2,1}\left ( L^2(\mathbb {R}^3) \right )$
. The linear equation
$$ \begin{align} i \varepsilon \partial_t \Gamma_t &= \left[ \left( - i \varepsilon \nabla - \eta * \boldsymbol{A}_{\alpha_t} \right)^2 + K * \rho_{\omega_t} - X_{\omega_t} , \Gamma_t \right] \end{align} $$
with initial datum
$\Gamma _0$
has a unique
$C^1 \left ( 0,T; \mathfrak {S}^{1} \left ( L^2(\mathbb {R}^3) \right ) \right ) \cap C \left ( 0,T; \mathfrak {S}^{2,1} \left ( L^2(\mathbb {R}^3) \right ) \right )$
solution.
Proof. Let
$(\mathcal {Z}_T, d_T)$
be the Banach space
$\mathcal {Z}_T = \left \{ \Gamma \in C \left ( 0,T; \mathfrak {S}^{2,1} \left ( L^2(\mathbb {R}^3) \right ) \right ) : \Gamma (0) = \Gamma _0 \right \}$
with metric
$d_T(\Gamma , \Gamma ') = \sup _{t \in [0,T]} \left \| \Gamma (t) - \Gamma '(t) \right \| _{\mathfrak {S}^{2,1}}$
. We define the mapping
$\Phi : \mathcal {Z}_T \rightarrow \mathcal {Z}_T$
by
$$ \begin{align} \Phi(\Gamma)(t) &= U_{\alpha}(t,0) \bigg( \Gamma_0 - i \varepsilon^{-1} \int_0^t U_{\alpha}(0,s) \left[ K * \rho_{\omega_s} - X_{\omega_s} , \Gamma_s \right] U_{\alpha}(s,0)\,ds \bigg) U_{\alpha}(0,t), \end{align} $$
where
$U_{\alpha }$
is defined as in Lemma 5.1. By means of (46a), (47b) and the properties of
$U_{\alpha }$
it is straightforward to see that the integrand on the right-hand side defines a strongly continuous function w.r.t. s on
$\mathfrak {S}^{1}$
and a summable function w.r.t. s on
$\mathfrak {S}^{2,1}$
, enabling us to define its integral on
$\mathfrak {S}^{1}$
as a Riemann integral and on
$\mathfrak {S}^{2,1}$
as a Bochner integral. Together with the strong differentiability of
$U_{\alpha }$
this proves that the right-hand side of (167) is an element of
$C^1 \left ( 0,T; \mathfrak {S}^{1} \left ( L^2(\mathbb {R}^3) \right ) \right ) \cap C \left ( 0,T; \mathfrak {S}^{2,1} \left ( L^2(\mathbb {R}^3) \right ) \right )$
, ensuring that the mapping
$\Phi $
is well defined. By means of (46a), (47b) and Lemma 5.1 we obtain
$$ \begin{align} d_T \left( \Phi(\Gamma), \Phi(\Gamma') \right) &\leq \varepsilon^{-1} \sup_{t \in [0,T]} \left\{ \int_0^t \left\| U_{\alpha}(t,s) \left[ K * \rho_{\omega_s} - X_{\omega_s} , \Gamma_s - \Gamma_s' \right] U_{\alpha}(s,t) \right\| _{\mathfrak{S}^{2,1}}ds \right\} \nonumber \\ &\leq d_T (\Gamma, \Gamma') T C_{\varepsilon} N^{-1} K_{\varepsilon, \alpha, T} \left\| \omega \right\| _{L_T^{\infty} \mathfrak{S}^{2,1}} \end{align} $$
for
$\Gamma , \Gamma ' \in \mathcal {Z}_T$
where
$C_{\varepsilon }$
is a generic constant depending on
$\varepsilon $
. Thus if we choose
$T^* \in (0,T]$
sufficiently small we obtain that
$\Phi $
is a contraction on
$(\mathcal {Z}_{T^*}, d_{T^*})$
. This proves the existence of a unique fixed point
$\Gamma \in \mathcal {Z}_{T^*}$
satisfying
$d_{T^*}(\Phi (\Gamma ), \Gamma ) = 0$
. By differentiation of (167) we obtain the existence of a unique solution to (166) until time
$T^*$
. The solution can be extended to the interval
$[0,T]$
because
$T^*$
can be chosen independently of
$\Gamma $
. Its regularity can be inferred from the regularity of the right-hand side of (167) as discussed above.
Lemma 5.3. Let
$T>0$
,
$(\omega , \alpha ) \in C \left ( 0,T; \mathfrak {S}^{1,1} \left ( L^2(\mathbb {R}^3 \right ) \right ) \times C(0,T; \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2})$
,
$\xi _0 \in \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2}$
and
$\boldsymbol {J}_{\omega _t, \alpha _t}$
be defined as in (10d). The linear equation
$$ \begin{align} i \partial_t \xi_t(k,\lambda) &= \left| k \right| \xi_t(k,\lambda) - \sqrt{ \frac{4 \pi^3}{\left| k \right|}} \mathcal{F}[\eta](k) \boldsymbol{\epsilon}_{\lambda}(k) \cdot \mathcal{F}[\boldsymbol{J}_{\omega_t, \alpha_t}](k) \end{align} $$
with initial datum
$\xi _0$
has a unique
$ C \big ( 0,T; \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2} \big ) \cap C^1 \big ( 0,T; \dot {\mathfrak {h}}_{-1/2} \big )$
solution.
Proof. We consider
$$ \begin{align} \xi_t(k,\lambda) &= e^{- i \left| k \right| t} \xi_0 (k,\lambda) + i e^{-i \left| k \right| t} \int_0^t e^{i \left| k \right| s} \sqrt{ \frac{4 \pi^3}{\left| k \right|}} \mathcal{F}[\eta](k) \boldsymbol{\epsilon}_{\lambda}(k) \mathcal{F}[\boldsymbol{J}_{\omega_s, \alpha_s}](k)\,ds. \end{align} $$
Note that the integrand on the right-hand side is a
$ C \big ( 0,T; \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2} \big )$
function because of (28), (68) and the regularity properties of
$(\omega ,\alpha )$
. If we define the integral in the Riemann sense, we obtain the strong differentiability of
$\xi _t$
with respect to time in
$\dot {\mathfrak {h}}_{-1/2}$
. Because (170) satisfies (169) this shows the existence of at least one solution of (169). In order to prove uniqueness let
$\xi $
and
$\xi '$
be two solutions of (169) with initial datum
$\xi _0$
. Since
$i \partial _t e^{i \left | k \right | t} \left ( \xi _t - \xi _t' \right ) = 0$
we get
$\xi = \xi '$
by Duhamel’s formula, showing the claim.
5.2 Local solutions
Lemma 5.4. For all
$(\omega _0, \alpha _0) \in \mathfrak {S}^{2,1} (L^2(\mathbb {R}^3)) \times \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2}$
there exists
$T>0$
and a unique
$C \big ( 0,T; \mathfrak {S}^{2,1} \big ( L^2(\mathbb {R}^3) \big ) \big ) \cap C^1 \big ( 0,T; \mathfrak {S}^{1} \big ( L^2(\mathbb {R}^3) \big ) \big ) \times C \big ( 0,T; \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2} \big ) \cap C^1 \big ( 0,T; \dot {\mathfrak {h}}_{-1/2} \big ) $
-valued function which satisfies (9) in
$\mathfrak {S}^1(L^2(\mathbb {R}^3)) \oplus \dot {\mathfrak {h}}_{-1/2}$
with initial datum
$(\omega _0, \alpha _0)$
.
Proof. The local existence of the solution will be shown by means of the contraction mapping principle. To this end, we define for
$(\omega _0, \alpha ) \in \mathfrak {S}^{2,1}(L^2(\mathbb {R}^3)) \times \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2}$
and
$T, R_1, R_2> 0$
satisfying
$R_2 \geq 1+ 2 \left \| \alpha _0 \right \| _{\mathfrak {h}_{1/2} \, \cap \, \dot {\mathfrak {h}}_{-1/2}}$
and
$R_1 \geq 1 + 20 K_{\varepsilon }^2 R_2^{8} \left \| \omega _0 \right \| _{\mathfrak {S}^{2,1}}$
(with
$K_\varepsilon $
being the constant of Lemma 5.1) the
$(T,R_1,R_2)$
-dependent space
$$ \begin{align} \mathcal{Z}_{T, R_1, R_2} &= \Big\{ (\omega, \alpha) \in \big( C(0,T; \mathfrak{S}^{1,1}(L^2(\mathbb{R}^3)) \cap L^{\infty}(0,T; \mathfrak{S}^{2,1}(L^2(\mathbb{R}^3)) \big) \nonumber \\ &\qquad \times \big( C(0,T; \mathfrak{h}_{1/2} \cap \dot{\mathfrak{h}}_{-1/2} ) \cap W^{1,2}(0,T; \dot{\mathfrak{h}}_{-1/2} ) \big) : (\omega(t),\alpha(t)) \big|_{t= 0} = (\omega_0, \alpha) , \nonumber \\ &\qquad \left\| \omega \right\| _{L_T^{\infty}\mathfrak{S}^{2,1}} \leq R_1 , \max \{ \left\| \alpha \right\| _{L_T^{\infty} \mathfrak{h}_{1/2} \, \cap \, L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}}, \left\| \partial_t \alpha \right\| _{L^2(0,T; \dot{\mathfrak{h}}_{-1/2} )} \} \leq R_2 \Big\}. \end{align} $$
Equipping
$\mathcal {Z}_{T, R_1, R_2}$
with the metric
$$ \begin{align} d_T \big( (\omega, \alpha) , (\omega', \alpha') \big) &= \max \left\{ \left\| \omega - \omega' \right\| _{L_T^{\infty}\mathfrak{S}^{1,1}} , \left\| \alpha - \alpha' \right\| _{L_T^{\infty} \mathfrak{h}_{1/2} \, \cap \, L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}} \right\}, \end{align} $$
where
$(\omega , \alpha )$
,
$(\omega ', \alpha ) \in \mathcal {Z}_{T, R_1, R_2}$
, leads to the Banach space
$(\mathcal {Z}_{T, R_1, R_2}, d_T)$
.
Next, we consider the solutions of the linearized equations from Lemma 5.2 and Lemma 6.3, satisfying
$$ \begin{align} \begin{cases} i \varepsilon \partial_t \Gamma_t \kern-7pt&= \left[ \left( - i \varepsilon \nabla - \eta * \boldsymbol{A}_{\alpha_t} \right)^2 + K * \rho_{\omega_t} - X_{\omega_t} , \Gamma_t \right] \\ i \partial_t \xi_t(k,\lambda) \kern-7pt&= \left| k \right| \xi_t(k,\lambda) - \sqrt{ \frac{4 \pi^3}{\left| k \right|}} \mathcal{F}[\eta](k) \boldsymbol{\epsilon}_{\lambda}(k) \cdot \mathcal{F}[\boldsymbol{J}_{\omega_t, \alpha_t}](k) \end{cases} , \end{align} $$
with initial datum
$(\omega _0, \alpha _0)$
, and define the mapping
$\Phi : \mathcal {Z}_{T, R_1, R_2} \rightarrow C^1 \left ( 0,T;\mathfrak {S}^{1} \left ( (L^2(\mathbb {R}^3) \right ) \right ) \cap C \left ( 0,T; \mathfrak {S}^{2,1} \left ( L^2(\mathbb {R}^3) \right ) \right ) \times C \big ( 0,T; \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2} \big ) \cap C^1 \big ( 0,T; \dot {\mathfrak {h}}_{-1/2} \big )$
by
$$ \begin{align} \Phi \begin{pmatrix} \omega \\ \alpha \end{pmatrix} &= \begin{pmatrix} \Gamma \\ \xi \end{pmatrix}. \end{align} $$
With this regard note that the assumptions on
$(\omega , \alpha )$
from Lemma 5.2 and Lemma 6.3 are satisfied because
$W^{1,2}(0,T; \dot {\mathfrak {h}}_{-1/2} ) \subset W^{1,1}(0,T; \dot {\mathfrak {h}}_{-1/2} )$
. Below, we will show that there exists a
$\varepsilon $
-dependent constant
$\widetilde {C}_{\varepsilon } \geq 0$
such that for all times
$T \leq \frac {1}{\widetilde {C}_{\varepsilon } R_1^2 R_2^8}$
the mapping
$\Phi : \mathcal {Z}_{T, R_1, R_2} \rightarrow \mathcal {Z}_{T, R_1, R_2}$
is well defined and is a contraction on
$(\mathcal {Z}_{T, R_1, R_2} , d_{T})$
. By the Banach fixed-point theorem we obtain the existence of a unique fixed point
$(\omega , \alpha ) \in \mathcal {Z}_{T, R_1, R_2}$
such that
$d \big ( (\omega , \alpha ) , \Phi (\omega , \alpha ) \big ) = 0$
. Together with the regularity properties of the linearized equations from Lemma 5.2 and Lemma 6.3 this proves Lemma 5.4.
Well-definedness of
$\Phi $
:
Next, we prove that, for
$T \leq \frac {1}{\widetilde {C}_{\varepsilon } R_1^2 R_2^8}$
and
$\widetilde {C}_{\varepsilon }>0$
chosen sufficiently large,
$\Phi $
maps
$\mathcal {Z}_{T, R_1, R_2}$
into itself. To this end we use the expansion (170) with
$\xi _0 = \alpha _0$
, (28) and (68) to estimate
$$ \begin{align} \left\| \xi_t \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} &\leq \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C \int_0^t \left\| \left< \cdot \right> \left| \cdot \right|{}^{-1} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \mathcal{F}[\boldsymbol{J}_{\omega_s, \alpha_s} \right\| _{L^{\infty}(\mathbb{R}^3, \mathbb{C}^3)}ds \nonumber \\ &\leq \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C_{\varepsilon} \int_0^t \left( 1+ \left\| \alpha_s \right\| _{\dot{\mathfrak{h}}_{-1/2}} \right) \left\| \omega_s \right\| _{\mathfrak{S}^{1,1}} ds \nonumber \\ &\leq \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C_{\varepsilon} \left( 1 + \left\| \alpha \right\| _{L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}} \right) \left\| \omega \right\| _{L_T^{\infty} \mathfrak{S}^{1,1}} t. \end{align} $$
Using
$R_2 \geq 1+ 2 \left \| \alpha _0 \right \| _{\mathfrak {h}_{1/2} \, \cap \, \dot {\mathfrak {h}}_{-1/2}}$
and
$(\omega , \alpha ) \in \mathcal {Z}_{T, R_1, R_2}$
we obtain
$$ \begin{align} \left\| \xi_t \right\| _{ L_T^{\infty} \mathfrak{h}_{1/2} \, \cap \, L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}} &\leq \frac{R_2}{2} + C_{\varepsilon} R_1 R_2 T \leq R_2 \end{align} $$
if
$\widetilde {C}_{\varepsilon }$
in the definition of T is chosen large enough. By similar means we get
$$ \begin{align} \left\| \partial_t \xi_t \right\| _{L_T^2 \dot{\mathfrak{h}}_{-1/2}} &\leq T^{1/2} \left\| \partial_t \xi_t \right\| _{L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}} \nonumber \\ &\leq T^{1/2} \left( \left\| \xi_t \right\| _{L_T^{\infty} \mathfrak{h}_{1/2}} + C \left\| \left| \cdot \right|{}^{-1} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \mathcal{F}[\boldsymbol{J}_{\omega_t, \alpha_t}] \right\| _{L_T^{\infty} L^{\infty}(\mathbb{R}^3, \mathbb{C}^3)} \right) \nonumber \\ &\leq T^{1/2} \left( \left\| \xi_t \right\| _{L_T^{\infty} \mathfrak{h}_{1/2}} + C_{\varepsilon} R_1 R_2 \right) \nonumber \\ &\leq R_2. \end{align} $$
To obtain the ultimate inequality we have used (176). Using the Duhamel expansion
$$ \begin{align} \Gamma_t &= U_{\alpha}(t;0) \omega_0 U^*_{\alpha}(t;0) - i \varepsilon^{-1} \int_0^t U_{\alpha}(t;s) \left[ K * \rho_{\omega_s} - X_{\omega_s} , \Gamma_s \right] U_{\alpha}^*(t;s)ds \end{align} $$
with
$U_{\alpha }$
being defined as in Lemma 5.1, (163) and Lemma 3.2 we estimate
$$ \begin{align} \left\| \Gamma \right\| _{L_t^{\infty} \mathfrak{S}^{2,1}} &\leq K_{ \varepsilon, \alpha, t}^2 \left( \left\| \omega_0 \right\| _{\mathfrak{S}^{2,1}} + C_{\varepsilon} \int_0^t \left\| \omega_s \right\| _{\mathfrak{S}^{2,1}} \left\| \Gamma_s \right\| _{\mathfrak{S}^{2,1}} ds\right) \nonumber \\ &\leq K_{\varepsilon, \alpha, T}^2 \left( \left\| \omega_0 \right\| _{\mathfrak{S}^{2,1}} + C_{\varepsilon} R_1 \int_0^t \left\| \Gamma \right\| _{L_s^{\infty} \mathfrak{S}^{2,1}}ds \right) \quad \text{for all} \; t \in [0,T]. \end{align} $$
Grönwall’s lemma then leads to
$$ \begin{align} \left\| \Gamma \right\| _{L_T^{\infty} \mathfrak{S}^{2,1}} &\leq K_{\varepsilon, \alpha, T}^2 \left\| \omega_0 \right\| _{\mathfrak{S}^{2,1}} \left( 1 + e^{C_{\varepsilon} R_1 K_{\alpha, \varepsilon, T}^2 T } \right). \end{align} $$
Note that
$$ \begin{align} K_{\varepsilon, \alpha, T} &\leq 2 K_{\varepsilon} R_2^4 e^{K_{\varepsilon} R_2 T^{1/2} } \leq 3 K_{\varepsilon} R_2^4 \end{align} $$
and
$$ \begin{align} \left\| \Gamma \right\| _{L_T^{\infty} \mathfrak{S}^{2,1}} &\leq \frac{R_1}{2} \left( 1 + e^{C_{\varepsilon} R_1 K_{\alpha, \varepsilon, T}^2 T } \right) \leq R_1 \end{align} $$
if we choose
$\widetilde {C}_{\varepsilon }$
in the definition of T sufficiently large because
$R_1 \geq 1 + 20 K_{\varepsilon }^2 R_2^{8} \left \| \omega _0 \right \| _{\mathfrak {S}^{2,1}}$
. In total, this shows that
$\Phi $
maps
$\mathcal {Z}_{T,R_1, R_2}$
into itself.
Contraction property of
$\Phi $
:
Next, we are going to prove that the mapping
$\Phi $
is a contraction on
$\mathcal {Z}_{T,R_1, R_2}$
. Let us consider
$(\omega ,\alpha )$
,
$(\omega ',\alpha ') \in \mathcal {Z}_{T,R_1,R_2}$
and denote their images under the mapping
$\Phi $
by
$(\Gamma , \xi )$
and
$(\Gamma ', \xi ')$
respectively. Using
$$ \begin{align} &i \varepsilon \partial_t U_{\alpha}^*(t;0) \left( \Gamma_t - \Gamma_t' \right) U_{\alpha}(t;0) \nonumber \\ &\quad = U_{\alpha}^*(t;0) \bigg( \left[ K * \rho_{\omega_t} - X_{\omega_t} , \Gamma_t \right] - \left[ K * \rho_{\omega_t'} - X_{\omega_t'} , \Gamma_t' \right] \nonumber \\ &\qquad - \left[ \left( - i \varepsilon \nabla - \eta * \boldsymbol{A}_{\alpha^{\prime}_t} \right)^2 - \left( - i \varepsilon \nabla - \eta * \boldsymbol{A}_{\alpha_t} \right)^2 , \Gamma_t' \right] \bigg) U_{\alpha}(t;0) \end{align} $$
and the fact that both solutions have the same initial condition, we get for
$t \in [0,T]$
$$ \begin{align} \left\| \Gamma_t - \Gamma_t' \right\| _{\mathfrak{S}^{1,1}} &\leq \varepsilon^{-1} \int_0^t \left\| U_{\alpha}(t;s) \left[ K * \rho_{\omega_s - \omega_s'} - X_{\omega_s - \omega_s'} , \Gamma_s \right] U_{\alpha}^*(t;s) \right\| _{\mathfrak{S}^{1,1}}ds \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \int_0^t \left\| U_{\alpha}(t;s) \left[ K * \rho_{\omega_s'} - X_{\omega_s'} , \left( \Gamma_s - \Gamma_s' \right) \right] U_{\alpha}^*(t;s) \right\| _{\mathfrak{S}^{1,1}}ds \end{align} $$
$$ \begin{align} &\quad + \varepsilon^{-1} \int_0^t \left\| U_{\alpha}(t;s) \left[ \big( (\eta * \boldsymbol{A}_{\alpha_s} )^2 - (\eta * \boldsymbol{A}_{\alpha_s'} )^2 \big) , \Gamma_s' \right] U_{\alpha}^*(t;s) \right\| _{\mathfrak{S}^{1,1}} ds \end{align} $$
$$ \begin{align} &\quad + 2 \int_0^t \left\| U_{\alpha}(t;s) \left[ \eta * \boldsymbol{A}_{\alpha_s - \alpha_s'} \cdot \nabla , \Gamma_s' \right] U_{\alpha}^*(t;s) \right\| _{\mathfrak{S}^{1,1}}ds. \end{align} $$
Next, we estimate each line separately. By means of Lemma 5.1 and Lemma 3.2 we obtain
$$ \begin{align} (184a) &\leq \varepsilon^{-1} \int_0^t \left\| U_{\alpha}(t;s) \right\| _{\mathfrak{S}^{\infty}(H^1(\mathbb{R}^3))} \left\| U_{\alpha}^*(t;s) \right\| _{\mathfrak{S}^{\infty}(H^1(\mathbb{R}^3))} \left\| \Gamma_s \right\| _{\mathfrak{S}^{1,1}} \nonumber \\ &\qquad \qquad \times \left( \left\| K * \rho_{\omega_s - \omega_s'} \right\| _{W_0^{1, \infty}(\mathbb{R}^3)} + C N^{-1} \left\| \omega_s - \omega_s' \right\| _{\mathfrak{S}^{1,1}} \right)ds \nonumber \\ &\leq C_{\varepsilon} K_{\varepsilon,\alpha,t} \, t \left\| \Gamma \right\| _{L_t^{\infty} \mathfrak{S}^{1,1}} \left\| \omega - \omega' \right\| _{L_t^{\infty} \mathfrak{S}^{1,1}} \end{align} $$
and
$$ \begin{align} (184b) &\leq C_{\varepsilon} K_{\varepsilon,\alpha,t} \, t \left\| \omega' \right\| _{L_t^{\infty} \mathfrak{S}^{1,1}} \left\| \Gamma - \Gamma' \right\| _{L_t^{\infty} \mathfrak{S}^{1,1}}. \end{align} $$
Using
$(\eta * \boldsymbol {A}_{\alpha _s} )^2 - (\eta * \boldsymbol {A}_{\alpha _s'} )^2 = \eta * \boldsymbol {A}_{\alpha _s - \alpha _s'} \, \eta * \boldsymbol {A}_{\alpha _s + \alpha _s'}$
, Lemma 5.1 and (40b) we get
$$ \begin{align} (184c) &\leq 2 \varepsilon^{-1} K_{\varepsilon,\alpha,t} \int_0^t \left\| \eta * \boldsymbol{A}_{\alpha_s - \alpha_s'} \right\| _{W_0^{1, \infty}(\mathbb{R}^3)} \left\| \eta * \boldsymbol{A}_{\alpha_s + \alpha_s'} \right\| _{W_0^{1, \infty}(\mathbb{R}^3)} \left\| \Gamma_s' \right\| _{\mathfrak{S}^{1,1}}ds \nonumber \\ &\leq C_{\varepsilon} K_{\varepsilon,\alpha,t} \, t \left( \left\| \alpha \right\| _{L_t^{\infty} \mathfrak{h}} + \left\| \alpha' \right\| _{L_t^{\infty} \mathfrak{h}} \right) \left\| \Gamma' \right\| _{L_t^{\infty} \mathfrak{S}^{1,1}} \left\| \alpha - \alpha' \right\| _{L_t^{\infty} \mathfrak{h}} \end{align} $$
and
$$ \begin{align} (184d) &\leq 2 K_{\varepsilon,\alpha,t} \int_0^t \left\| \eta * \boldsymbol{A}_{\alpha - \alpha'} \right\| _{W_0^{1, \infty}(\mathbb{R}^3)} \left( \left\| \nabla \Gamma_s' \right\| _{\mathfrak{S}^{1,1}} + \left\| \Gamma_s' \nabla \right\| _{\mathfrak{S}^{1,1}} \right)ds \nonumber \\ &\leq C K_{\varepsilon,\alpha,t} \, t \left\| \Gamma \right\| _{L_t^{\infty} \mathfrak{S}^{2,1}} \left\| \alpha - \alpha' \right\| _{L_t^{\infty} \mathfrak{h}}. \end{align} $$
Summing all contributions together, taking the supremum in t over the interval
$[0,T]$
, and using that
$(\omega , \alpha )$
,
$(\omega ', \alpha )$
,
$(\Gamma , \xi )$
,
$(\Gamma ', \xi ') \in \mathcal {Z}_{T,R_1,R_2}$
lead to
$$ \begin{align} \left\| \Gamma - \Gamma' \right\| _{L_T^{\infty} \mathfrak{S}^{1,1}} &\leq C K_{\varepsilon, \alpha, T} R_1 \, T \left( \left\| \Gamma - \Gamma' \right\| _{L_T^{\infty} \mathfrak{S}^{1,1}} + R_2 \, d_T \left( (\omega, \alpha) , (\omega', \alpha') \right) \right). \end{align} $$
By means of (181) we have
$C K_{\varepsilon , \alpha , T} R_1 \, T \leq 1/2$
and
$2 C K_{\varepsilon , \alpha , T} R_1 R_2 \, T < 1$
for
$\widetilde {C}_\varepsilon $
in the definition of T chosen sufficiently large. This leads to
$$ \begin{align} \left\| \Gamma - \Gamma' \right\| _{L_T^{\infty} \mathfrak{S}^{1,1}} &< d_T \left( (\omega, \alpha) , (\omega', \alpha') \right). \end{align} $$
Using Duhamel’s formula, (28) and (68) we get
$$ \begin{align} \left\| \xi_t - \xi_t' \right\| _{L_T^{\infty} \mathfrak{h}_{1/2} \, \cap \, L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}} &\leq C \sup_{t \in [0,T]} \int_0^t \left\| \left< \cdot \right> \left| \cdot \right|{}^{-1} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \mathcal{F}[\boldsymbol{J}_{\omega_s, \alpha_s} - \boldsymbol{J}_{\omega_s', \alpha_s'}] \right\| _{L^{\infty}(\mathbb{R}^3, \mathbb{C}^3)}ds \nonumber \\ &\leq C_{\varepsilon} \left( 1 + \left\| \omega' \right\| _{L_T^{\infty} \mathfrak{S}^1} + \left\| \alpha \right\| _{L_T^{\infty} \mathfrak{h}} \right) T d_T \left( (\omega, \alpha) , (\omega', \alpha') \right) \nonumber \\ &\leq C_{\varepsilon} \left( 1 + R_1 + R_2 \right) T d_T \left( (\omega, \alpha) , (\omega', \alpha') \right) \nonumber \\ &< d_T \left( (\omega, \alpha) , (\omega', \alpha') \right). \end{align} $$
For
$\widetilde {C}_{\varepsilon }>0$
chosen sufficiently large and
$T \leq \frac {1}{\widetilde {C}_{\varepsilon } R_1^2 R_2^8}$
the estimates above imply
$$ \begin{align} d_{T} \big( \Phi (\omega, \alpha), \Phi (\omega' , \alpha' ) \big) < d_{T} \big( (\omega, \alpha) , (\omega' , \alpha' ) \big) , \end{align} $$
showing that
$\Phi $
is a contraction on
$\mathcal {Z}_{T,R_1,R_2}$
.
5.3 Global solutions
Here below we conclude the proof of Proposition 2.1 concerning the well-posedness and regularity theory for the Vlasov–Maxwell system.
Proof of Proposition 2.1.
From Lemma 5.4 we infer the existence of a time T and a unique local
$C \big ( 0,T; \mathfrak {S}^{2,1} \big ( L^2(\mathbb {R}^3) \big ) \big ) \cap C^1 \big ( 0,T; \mathfrak {S}^{1} \big ( L^2(\mathbb {R}^3) \big ) \big ) \times C \big ( 0,T; \mathfrak {h}_{1/2} \cap \dot {\mathfrak {h}}_{-1/2} \big ) \cap C^1 \big ( 0,T; \dot {\mathfrak {h}}_{-1/2} \big ) $
-valued solution of (9) with initial datum
$(\omega _0, \alpha _0)$
, which will be denoted by
$(\omega , \alpha )$
in the following. Let
$t \in [0,T]$
,
$H_{\omega ,\alpha }(t) = \left ( - i \varepsilon \nabla - \eta * \boldsymbol {A}_{\alpha _t} \right )^2 + K * \rho _{\omega _t} - X_{\omega _t}$
and
$\psi \in H^2(\mathbb {R}^3)$
. Using (39a), (46a), (47a) and the regularity properties of
$(\omega ,\alpha )$
it is straightforward to show the estimate
$$ \begin{align} \left\| \left( H_{\omega, \alpha}(t) + \varepsilon^2 \Delta \right) \psi \right\| _{L^2(\mathbb{R}^3)} &\leq \delta \left\| - \varepsilon^2 \Delta \psi \right\| _{L^2(\mathbb{R}^3)} + C \Big[ \left( 1 + \delta^{-1} \right) \left\| \alpha_t \right\| _{\dot{\mathfrak{h}}_{-1/2}}^2 \nonumber \\ &\quad + N^{-1} \left\| \omega_t \right\| _{\mathfrak{S}^1} \Big] \left\| \psi \right\| _{L^2(\mathbb{R}^3)} \end{align} $$
and the continuous differentiability of the mapping
$t\in [0,T] \mapsto H_{\omega , \alpha }(t) \psi \in L^2(\mathbb {R}^3)$
. Inequality (193) and the Kato-Rellich theorem imply that
$H_{\omega ,\alpha }(t)$
is a self-adjoint operator with domain
$\mathcal {D}( H_{\omega , \alpha }(t)) = H^2(\mathbb {R}^3)$
. Together with the strong differentiability this (see [Reference Reed and Simon48, Theorem X.70, proof of Theorem X.71] and [Reference Griesemer and Schmid27, Theorem 2.2]) gives rise to a two-parameter family
$\{ U_{\omega ,\alpha }(t,s) \}_{(t,s) \in [0,T]^2}$
of unitary operators on
$L^2(\mathbb {R}^3)$
such that
$ U_{\omega ,\alpha }(t;s) H^2(\mathbb {R}^3) \subset H^2(\mathbb {R}^3)$
and
$\psi _s(t) = U_{\omega ;\alpha }(t,s) \psi $
with
$\psi \in H^2(\mathbb {R}^3)$
is strongly continuous differentiable and satisfies
$\frac {d}{dt} \psi _s(t) = - i \varepsilon ^{-1} H_{\omega , \alpha }(t) \psi _s(t)$
,
$\psi _s(s) = \psi $
. The operator
$U_{\omega , \alpha }(t;0) \omega _0 U_{\omega , \alpha }(t;0)^* \in C \big ( 0,T; \mathfrak {S}^{2,1} \big ( L^2(\mathbb {R}^3) \big ) \big ) \cap C^1 \big ( 0,T; \mathfrak {S}^{1} \big ( L^2(\mathbb {R}^3) \big ) \big )$
satisfies the first equation of (9) with initial value
$\omega _0$
. Uniqueness then implies that
$\omega _t = U_{\omega , \alpha }(t;0) \omega _0 U_{\omega , \alpha }(t;0)^*$
, hence
$\omega _t \in \mathfrak {S}_{+}^{1}$
for all
$t \in [0,T]$
because
$\omega _0 \geq 0$
by assumption. The conservation of mass and energy, that is,
$$ \begin{align} \text{Tr} \left( \omega_t \right) = \text{Tr} \left( \omega_0 \right) \quad \text{and} \quad \mathcal{E}^{\mathrm{{MS}}}[\omega_t, \alpha_t] = \mathcal{E}^{\mathrm{{MS}}}[\omega_0, \alpha_0] \quad \text{for all} \; t \in [0,T] , \end{align} $$
are obtained by direct inspection. In the following, we will prove
$$ \begin{align} \left\| \alpha_t \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} &\leq \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + t \, C \big( \varepsilon, N \left\| \omega_0 \right\| _{\mathfrak{S}^{1,1}} , \left\| \alpha_0 \right\| _{\dot{\mathfrak{h}}_{1/2}} \big) \end{align} $$
and
$$ \begin{align} \left\| \omega_t \right\| _{\mathfrak{S}^{2,1}} &\leq \exp \left[ \exp \left[ \left< t \right>^2 \, C \big( \varepsilon, N \left\| \omega_0 \right\| _{\mathfrak{S}^{2,1}} , \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} \big) \right] \right] \end{align} $$
for all
$t \in [0,T]$
. These estimates lead to the global existence of solutions because, by standard methods (see, e.g., [Reference Cazenave and Haraux15, Theorem 4.3.4]), one can derive from the contraction mapping principle a blow-up alternative which states that the maximal time of existence
$T_{\text {max}}$
is either infinite or
$\lim _{t \nearrow T_{\text {max}}} ( \left \| \omega _t \right \| _{\mathfrak {S}^{2,1}} + \left \| \alpha _t \right \| _{\mathfrak {h}_{1/2} \, \cap \, \dot {\mathfrak {h}}_{-1/2}} ) = \infty $
.
Inequality (195).
Using (69a)–(69c) and the conservation of mass and energy, we get
$$ \begin{align} \left\| \omega_t \right\| _{\mathfrak{S}^{1,1}} + \left\| \alpha_t \right\| _{\dot{\mathfrak{h}}_{1/2}}^2 &\leq C \big( \varepsilon, N \left\| \omega_0 \right\| _{\mathfrak{S}^{1,1}} , \left\| \alpha_0 \right\| _{\dot{\mathfrak{h}}_{1/2}} \big) \quad \text{for all} \; t \in [0,T]. \end{align} $$
Together with Duhamel’s formula, (28) and (68) this leads to
$$ \begin{align} \left\| \alpha_t \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} &\leq \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C \int_0^t \left\| \left< \cdot \right> \left| \cdot \right|{}^{-1} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \mathcal{F}[\boldsymbol{J}_{\omega_s, \alpha_s}] \right\| _{L^{\infty}(\mathbb{R}^3, \mathbb{C}^3)}ds \nonumber \\ &\leq \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C_{\varepsilon} \int_0^t \left( 1 + \left\| \alpha_s \right\| _{\dot{\mathfrak{h}}_{1/2}} \right) \left\| \omega_s \right\| _{\mathfrak{S}^{1,1}}ds \nonumber \\ &\leq \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + t \, C \big( \varepsilon, N \left\| \omega_0 \right\| _{\mathfrak{S}^{1,1}} , \left\| \alpha_0 \right\| _{\dot{\mathfrak{h}}_{1/2}} \big) \quad \text{for all} \; t \in [0,T]. \end{align} $$
Inequality (196).
Using
$U_{\alpha }$
, defined as in Lemma 5.1, we get the Duhamel expansion
$$ \begin{align} \omega_t &= U_{\alpha}(t;0) \omega_0 U_{\alpha}^*(t;0) - i \varepsilon^{-1} \int_0^t U_{\alpha}(t;s) \left[ K * \rho_{\omega_s} - X_{\omega_s} , \omega_s \right] U_{\alpha}^*(t;s)\,ds. \end{align} $$
Lemma 5.1 and (46a) then lead to
$$ \begin{align} \left\| \omega_t \right\| _{\mathfrak{S}^{2,1}} &\leq K_{\varepsilon, \alpha, t}^2 \left( \left\| \omega_0 \right\| _{\mathfrak{S}^{2,1}} + C_{\varepsilon} \int_0^t \left( \left\| \omega_s \right\| _{\mathfrak{S}^{1}} \left\| \omega_s \right\| _{\mathfrak{S}^{2,1}} + \left\| \left[ X_{\omega_s}, \omega_s \right] \right\| _{\mathfrak{S}^{2,1}} \right)\,ds \right). \end{align} $$
Since
$\omega _s \in \mathfrak {S}_{+}^{2,1}(L^2(\mathbb {R}^3))$
, there exists a spectral set
$\{ \lambda _j, \psi _j \}_{j \in \mathbb {N}}$
with
$\lambda _j \geq 0$
for all
$j \in \mathbb {N}$
such that
$\omega _s = \sum _{j \in \mathbb {N}} \lambda _j | \psi _j \rangle \langle \psi _j |$
,
$\left \| \omega _s \right \| _{\mathfrak {S}^{1}} = \sum _{j \in \mathbb {N}} \lambda _j \left \| \psi _j \right \| _{L^2(\mathbb {R}^3)}$
and
$\left \| \omega _s \right \| _{\mathfrak {S}^{2,1}} = \sum _{j \in \mathbb {N}} \lambda _j \left \| \psi _j \right \| _{H^2(\mathbb {R}^3)}$
. This allows us to write out the expression involving the exchange term quite explicitly. More concretely, we bound
$$ \begin{align} \left\| \left[ X_{\omega_s}, \omega_s \right] \right\| _{\mathfrak{S}^{2,1}} &\leq \left\| X_{\omega_s} \omega_s \right\| _{\mathfrak{S}^{2,1}} + \left\| \omega_s X_{\omega_s} \right\| _{\mathfrak{S}^{2,1}} \end{align} $$
and then use the projection property of
$| \psi _j \rangle \langle \psi _j |$
to estimate
$$ \begin{align} \left\| X_{\omega_s} \omega_s \right\| _{\mathfrak{S}^{2,1}} &\leq \sum_{j \in \mathbb{N}} \lambda_j \left\| \left( 1 - \Delta \right) X_{\omega_s} | \psi_j \rangle \langle \psi_j | \left( 1 - \Delta \right) \right\| _{\mathfrak{S}^1} \nonumber \\ &\leq \sum_{j \in \mathbb{N}} \lambda_j \left\| \left( 1 - \Delta \right) X_{\omega_s} | \psi_j \rangle \langle \psi_j | \right\| _{\mathfrak{S}^2} \, \left\| | \psi_j \rangle \langle \psi_j |\left( 1 - \Delta \right) \right\| _{\mathfrak{S}^2} \nonumber \\ &= \sum_{j \in \mathbb{N}} \lambda_j \left\| \psi_j \right\| _{H^2} \left\| (1 - \Delta) X_{\omega_s} \psi_j \right\| _{L^2(\mathbb{R}^3)} \nonumber \\ &\leq N^{-1} \sum_{j,k \in \mathbb{N}} \lambda_j \lambda_k \left\| \psi_j \right\| _{H^2(\mathbb{R}^3)} \left\| \psi_k K * \{ \overline{\psi_k} \psi_j \} \right\| _{H^2(\mathbb{R}^3)} \nonumber \\ &\leq N^{-1} \sum_{j,k \in \mathbb{N}} \lambda_j \lambda_k \left\| \psi_j \right\| _{H^2(\mathbb{R}^3)} \left\| \psi_k \right\| _{H^2(\mathbb{R}^3)} \left\| K * \{ \overline{\psi_k} \psi_j \} \right\| _{W_0^{2,\infty}(\mathbb{R}^3)}. \end{align} $$
Using that
$\left \| \left < \cdot \right>^2 \mathcal {F}[K] \right \| _{L^1(\mathbb {R}^3)} \leq C$
because of (48) and (28), Young’s inequality and the Cauchy–Schwarz inequality, we obtain
$$ \begin{align} \left\| X_{\omega_s} \omega_s \right\| _{\mathfrak{S}^{2,1}} &\leq N^{-1} \left\| K \right\| _{W_0^{2,\infty}(\mathbb{R}^3)} \sum_{j,k \in \mathbb{N}} \lambda_j \lambda_k \left\| \psi_j \right\| _{H^2(\mathbb{R}^3)} \left\| \psi_k \right\| _{H^2(\mathbb{R}^3)} \left\| \overline{\psi_k} \psi_j \right\| _{L^1(\mathbb{R}^3)} \nonumber \\ &\leq N^{-1} \left\| \left< \cdot \right>^2 \mathcal{F}[K] \right\| _{L^1(\mathbb{R}^3)} \sum_{j,k \in \mathbb{N}} \lambda_j \lambda_k \left\| \psi_j \right\| _{H^2(\mathbb{R}^3)} \left\| \psi_j \right\| _{L^2(\mathbb{R}^3)} \left\| \psi_k \right\| _{H^2(\mathbb{R}^3)} \left\| \psi_k \right\| _{L^2(\mathbb{R}^3)} \nonumber \\ &\leq C N^{-1} \left( \sum_{j \in \mathbb{N}} \lambda_j \left\| \psi_j \right\| _{H^2(\mathbb{R}^3)}^2 \right) \left( \sum_{k \in \mathbb{N}} \lambda_k \left\| \psi_k \right\| _{L^2(\mathbb{R}^3)}^2 \right) \nonumber \\ &\leq C N^{-1} \left\| \omega_s \right\| _{\mathfrak{S}^{1}} \left\| \omega_s \right\| _{\mathfrak{S}^{2,1}}. \end{align} $$
Plugging this into (200) and the conservation of mass lead to
$$ \begin{align} \left\| \omega_t \right\| _{\mathfrak{S}^{2,1}} &\leq K_{\varepsilon, \alpha, t}^2 \left( \left\| \omega_0 \right\| _{\mathfrak{S}^{2,1}} + C_{\varepsilon} \left\| \omega_0 \right\| _{\mathfrak{S}^{1}} \int_0^t \left\| \omega_s \right\| _{\mathfrak{S}^{2,1}}ds \right). \end{align} $$
Combining similar estimates as in the proof of (195) with (195) gives
$$ \begin{align} \left\| \partial_t \alpha_t \right\| _{\dot{\mathfrak{h}}_{- 1/2}} &\leq \left\| \alpha_t \right\| _{\dot{\mathfrak{h}}_{1/2}} + C \int_0^t \left\| \left| \cdot \right|{}^{-1} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \mathcal{F}[\boldsymbol{J}_{\omega_s, \alpha_s}] \right\| _{L^{\infty}(\mathbb{R}^3, \mathbb{C}^3)}ds \nonumber \\ &\leq \left< t \right> \, C \big( \varepsilon, N \left\| \omega_0 \right\| _{\mathfrak{S}^{1,1}} , \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} \big) \quad \text{for all} \; t \in [0,T]. \end{align} $$
Inserting the previous estimate and (195) into the right-hand side of (163) yields
$$ \begin{align} K_{\varepsilon, \alpha, t} &\leq e^{\left< t \right>^2 \, C \big( \varepsilon, N \left\| \omega_0 \right\| _{\mathfrak{S}^{1,1}} , \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} \big)} , \end{align} $$
leading to
$$ \begin{align} \left\| \omega_t \right\| _{\mathfrak{S}^{2,1}} &\leq e^{\left< t \right>^2 \, C \big( \varepsilon, N \left\| \omega_0 \right\| _{\mathfrak{S}^{2,1}} , \left\| \alpha_0 \right\| _{\mathfrak{h}_{1/2} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} \big)} \left( 1 + \int_0^t ds \, \left\| \omega_s \right\| _{\mathfrak{S}^{2,1}} \right). \end{align} $$
Application of Grönwall’s Lemma then gives (196).
6 Solution theory of the regularized Vlasov–Maxwell system
In this section, we prove the existence of unique global solutions for (15). Similarly as in [Reference Degond22] we restrict our consideration to initial data with compact support in the velocity variable.
6.1 Propagation estimates for the characteristics
Lemma 6.1. Let
$T>0$
,
$(x,v) \in \mathbb {R}^6$
,
$f \in C(0,T; W_4^{0,2}(\mathbb {R}^6))$
and
$\alpha \in C(0,T; \mathfrak {h}_1)$
. Then,
$$ \begin{align} \begin{cases} \dot{X}_t(x,v)\kern-7pt &= 2 V_t(x,v) - 2 \eta * \boldsymbol{A}_{\alpha_t}(X_t(x,v)) \\ \dot{V}_t(x,v) \kern-7pt&= - \boldsymbol{F}_{f_t, \alpha_t} ( X_t(x,v), V_t(x,v)) \end{cases} \end{align} $$
with initial datum
$(X_t(x,v), V_t(x,v)) \big |_{t=0} = (x,v)$
has a unique
$C^1 \big ( 0,T, \mathbb {R}^6 \big )$
solution satisfying
$$ \begin{align} \left< V_t(x,v) \right> &\leq C \left< v \right> e^{C \int_0^t \left( \left\| f_s \right\| _{W_4^{0,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}}^2 \right)ds} , \end{align} $$
$$ \begin{align} \left<X_t(x,v) \right>&\leq C \left<(x,v) \right> \left< t \right> e^{C \int_0^t \left( \left\| f_s \right\| _{W_4^{0,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}}^2 \right)ds}. \end{align} $$
Proof. The existence of a unique local solution can be shown by a standard fixed point argument. Using (40b) and (46b) we obtain
$$ \begin{align} \sup_{x \in \mathbb{R}^3} \left| \boldsymbol{F}_{f_s, \alpha_s}(x,v) \right| &\leq C \left< v \right> \left( \left\| f_s \right\| _{W_4^{0,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}}^2 \right). \end{align} $$
Together with (208) this shows
$$ \begin{align} \left| V_t(x,v) \right| &\leq v + C \int_0^t \left( \left\| f_s \right\| _{W_4^{0,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}}^2 \right) \left< V_s(x,v) \right> ds \end{align} $$
and (209a) by Grönwall’s lemma. Inequality (209b) is a consequence of (208), (40b) and (209a). By means of (209a) and (209b) the solution can then be extended to the whole interval
$[0,T]$
.
6.2 Linear equations
Lemma 6.2. Let
$R>0$
,
$T>0$
,
$a, b \in \mathbb {N}$
such that
$a \geq 4$
and
$b \geq 3$
. Moreover, let
$(f, \alpha ) \in C \big ( 0,T; W_a^{b-1,2}(\mathbb {R}^6) \big ) \times C \big ( 0,T; \mathfrak {h}_{b} \cap \dot {\mathfrak {h}}_{-1/2} \big )$
and
$g_0 \in W_{a}^{b,2}(\mathbb {R}^6)$
such that
$\operatorname {\mathrm {supp}} g_0 \subset A_R$
with
$A_R = \{ (x,v) \in \mathbb {R}^6 , \left | v \right | \leq R \}$
. Then,
$$ \begin{align} \partial_t g_t &= - 2 \left( v - \eta * \boldsymbol{A}_{\alpha_t} \right) \cdot \nabla_x g_t + \boldsymbol{F}_{f_t, \alpha_t} \cdot \nabla_v g_t \end{align} $$
with initial datum
$g_0$
and
$\boldsymbol {F}_{f_t, \alpha _t} $
being defined as in (16b) has a unique
$L^{\infty } \big ( 0,T; W_{a}^{b,2}(\mathbb {R}^6) \big ) \cap C \big ( 0,T; W_{a}^{b-1,2}(\mathbb {R}^6) \big ) \cap C^1 (0,T; W_{a}^{b-2,2}(\mathbb {R}^6))$
solution. In addition,
$\operatorname {\mathrm {supp}} g_t \subset A_{R(t)}$
with
$R(t) \leq C \left < R \right> \exp \left ( C \int _0^t \big ( \left \| f_s \right \| _{W_4^{0,2}(\mathbb {R}^6)} + \left \| \alpha _s \right \| _{ \mathfrak {h}}^2 \big )ds \right )$
and
$$ \begin{align} \left\| g_t \right\| _{W_a^{b,2}(\mathbb{R}^6)} &\leq C \left\| g_0 \right\| _{W_a^{b,2}} \exp \left( C \int_0^t R(s) \big( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \big)ds \right). \end{align} $$
Proof. For
$T,R_1, R_2> 0$
we consider the Banach space
$(\mathcal {Z}_{T,R_1,R_2}, d_T)$
with
and metric
$d_T(g,g') = \left \| g - g' \right \| _{L_T^{\infty } W_a^{b-1,2}(\mathbb {R}^6)}$
. We, moreover, define the mapping
$\Phi : \mathcal {Z}_{T,R_1,R_2} \rightarrow \mathcal {Z}_{T,R_1,R_2}$
by
$$ \begin{align} \Phi_t(g) = g_0 + \int_0^t \left( - 2 \left( v - \eta * \boldsymbol{A}_{\alpha_s} \right) \cdot \nabla_x g_s + \boldsymbol{F}_{f_s, \alpha_s} \cdot \nabla_v g_s \right) ds. \end{align} $$
For all
$$ \begin{align} R_2 \geq C \left\| g_0 \right\| _{W_a^{b,2}(\mathbb{R}^6)} \quad \text{and} \quad T^* \leq \frac{1}{C \left< R_1 \right> \left( 1 + \left\| f \right\| _{L_T^{\infty} W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha \right\| _{L_T^{\infty} \mathfrak{h}_b}^2 \right) } \end{align} $$
with C chosen sufficiently large it is shown below that the mapping
$\Phi $
is well-defined and a contraction on
$(\mathcal {Z}_{T^*,R_1,R_2},d_{T^*})$
. By the Banach fixed point theorem this proves the existence of a unique
$g \in C (0,T^*; W_{a}^{b-1,2}(\mathbb {R}^6)) \cap L^{\infty }(0,T^*; W_a^{b,2}(\mathbb {R}^6))$
satisfying
$$ \begin{align} g_t = g_0 + \int_0^t \left( - 2 \left( v - \eta * \boldsymbol{A}_{\alpha_s} \right) \cdot \nabla_x g_s + \boldsymbol{F}_{f_s, \alpha_s} \cdot \nabla_v g_s \right) ds \end{align} $$
in
$W_{a}^{b-1,2}(\mathbb {R}^6))$
. Note that
$\eta * \boldsymbol {A}_{\alpha } \in C(0,T; W_0^{b+1,\infty }(\mathbb {R}^3))$
and
$K * \widetilde {\rho }_{f} \in C(0,T; W_0^{b+2,\infty }(\mathbb {R}^3))$
because of (40b) and (46b) and the assumed regularity of
$(f,\alpha )$
. This allows us to conclude that the integrand in the equation above is an element of
$C (0,T^*; W_{a}^{b-2,2}(\mathbb {R}^6))$
. Defining the integral in the Riemann sense then proves
$\Phi (g) \in C^1 (0,T^*; W_{a}^{b-2,2}(\mathbb {R}^6))$
. Let
$\big ( X^{-1}_{t}(x,v) , V^{-1}_{t}(x,v)) \big )$
be the backward flow of the characteristics (208) with initial datum
$(X_t^{-1}(x,v), V_t^{-1}(x,v)) \big |_{t=0} = (x,v)$
. Since
$$ \begin{align} &\partial_t g_0 \big( X_t^{-1}(x,v) , V_t^{-1}(x,v)) \big) \nonumber \\ &\quad = - \left( 2 V_t^{-1}(x,v) - 2 \eta * \boldsymbol{A}_{\alpha_t}(X_t^{-1}(x,v)) \right) \cdot \left( \nabla_1 g_0 \right) \big( X_t^{-1}(x,v) , V_t^{-1}(x,v) \big) \nonumber \\ &\qquad + \boldsymbol{F}_{f_t, \alpha_t} ( X_t^{-1}(x,v), V_t^{-1}(x,v)) \cdot \left( \nabla_2 g_0 \right) \big( X_t^{-1}(x,v) , V_t^{-1}(x,v) \big) \end{align} $$
we obtain
$g_t(x,v) = g_0 \big ( X_t^{-1}(x,v) , V_t^{-1}(x,v)) \big )$
for all
$t \in [0, T^*]$
by the uniqueness of (212). Using the support assumption on
$g_0$
, the fact that
$(x,v) = \big ( X_t \big ( X_t^{-1}(x,v) \big ) , V_t \big ( V_t^{-1}(x,v) \big ) \big )$
and the propagation estimate (209a) we get that
$\operatorname {\mathrm {supp}} g_t \subset A_{R(t)}$
with
$$ \begin{align} R(t) &\leq C \left< R \right> e^{ C \int_0^t \big( \left\| f_s \right\| _{W_4^{0,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{ \mathfrak{h}}^2 \big)ds }. \end{align} $$
This allows us to choose
$R_1 = 2 C \left < R \right> \exp \big ( C T \big ( \left \| f_s \right \| _{L_T^{\infty } W_4^{0,2}(\mathbb {R}^6)} + \left \| \alpha \right \| _{L_T^{\infty } \mathfrak {h}}^2 \big ) \big )$
in the contraction argument below. In this case
$T>0$
is fixed,
$R_1$
is independent of t and
$\operatorname {\mathrm {supp}}{g_t} \subset A_{R_1/2}$
holds for all t in the time interval of existence. We can therefore patch local solutions together until we obtain a solution on the whole interval
$[0,T]$
.
Proof of well-definedness.
Since
$g \in L^{\infty } (0,T; W_a^{b,2}(\mathbb {R}^6))$
we can define the integral in the definition of
$\Phi $
on
$W_{a-1}^{b-1,2}(\mathbb {R}^6)$
as a Bochner integral, implying that
$\Phi (g) \in C (0,T; W_{a-1}^{b-1,2}(\mathbb {R}^6))$
. Using
$L^{\infty } (0,T; W_a^{b,2}(\mathbb {R}^6))$
with
$b \geq 3$
we have that
$g_t \in C^1(\mathbb {R}^6)$
for all
$t \in [0,T]$
and therefore that 
holds pointwise for all
$t \in [0,T]$
. This implies that 
for all
$t \in [0,T]$
and therefore 
. Using
$g \in L^{\infty } (0,T; W_a^{b,2}(\mathbb {R}^6))$
again it then follows that
$\Phi (g) \in C (0,T; W_{a}^{b-1,2}(\mathbb {R}^6))$
. Next, we derive an estimate for
$\left \| \left ( 1 - \Delta _z \right )^{b/2} \left < \cdot \right>^a \Phi _t(g) \right \| _{L^2(\mathbb {R}^6)}$
. To this end let us define the regularized Laplacian
$$ \begin{align} \left( 1 - \Delta_z \right)_{\leq n} = \frac{\left( 1 - \Delta_z \right)}{\left( 1 - \Delta_z / n^2 \right)} \quad \text{with} \quad n \in \mathbb{N} \end{align} $$
and the transport operator
$$ \begin{align} T_{f, \alpha} = 2 \left( v - \eta * \boldsymbol{A}_{\alpha} \right) \cdot \nabla_x - \boldsymbol{F}_{f, \alpha} \cdot \nabla_v. \end{align} $$
By means of
$$ \begin{align} \Phi_t(g) &= g_0 - \int_0^t T_{f_s, \alpha_s}[g_s]\,ds \end{align} $$
we compute
$$ \begin{align} &\left\| \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a \Phi_t(g) \right\| _{L^2(\mathbb{R}^6)}^2 - \left\| \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a g_0 \right\| _{L^2(\mathbb{R}^6)}^2 \nonumber \\ &\quad = - 2 \int_0^t \left\langle \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a g_s , \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a T_{f_s, \alpha_s} [g_s] \right\rangle ds \nonumber \\ &\quad = - 2 \int_0^t \left\langle \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a g_s , T_{f_s, \alpha_s} \Big[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a g_s \Big] \right\rangle ds \nonumber \\ &\qquad - 2 \int_0^t \left\langle \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a g_s , \Big[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a , T_{f_s, \alpha_s} \Big] [g_s] \right\rangle ds. \end{align} $$
Note that
$\left [ F_{f, \alpha } , \cdot \nabla _v \right ] = 0$
because we are working in the Coulomb gauge and integration by parts leads to
$$ \begin{align} 2 \int_{\mathbb{R}^6} m(x,v) n(x,v) T_{f , \alpha}[n](x,v) dx\,dv &= - \int_{\mathbb{R}^6} T_{f , \alpha}[m](x,v) n^2(x,v) dx\,dv \end{align} $$
for sufficiently regular functions
$m, n: \mathbb {R}^6 \rightarrow \mathbb {R}$
. This lets us conclude
$$ \begin{align} \left\langle \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a g_s , T_{f_s, \alpha_s} \Big[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a g_s \Big] \right\rangle = 0 , \end{align} $$
leading to
$$ \begin{align} &\left\| \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a \Phi_t(g) \right\| _{L^2(\mathbb{R}^6)}^2 - \left\| \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a g_0 \right\| _{L^2(\mathbb{R}^6)}^2 \nonumber \\ &\quad = - 2 \int_0^t \left\langle \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a g_s , \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \Big[ \left< \cdot \right>^a , T_{f_s, \alpha_s} \Big] [g_s] \right\rangle ds \end{align} $$
$$ \begin{align} &\qquad - 2 \int_0^t \left\langle \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a g_s , \Big[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} , T_{f_s, \alpha_s} \Big] [ \left< \cdot \right>^a g_s ] \right\rangle ds. \end{align} $$
Note that
$$ \begin{align} \left[ T_{f_s, \alpha_s} , \left< z \right>^a \right] &= a \left< z \right>^{a -2 } \left( 2 \left( v - \eta * \boldsymbol{A}_{\alpha_s} \right) \cdot x - \boldsymbol{F}_{f_s, \alpha_s} \cdot v \right) \end{align} $$
because
$$ \begin{align} \nabla_x \left< z \right>^{a} &= a \left< z \right>^{a -2} x \quad \text{and} \quad \nabla_v \left< z \right>^{a} = a \left< z \right>^{a -2} v. \end{align} $$
If we use the Cauchy-Schwarz inequality and
$\left ( 1 - \Delta _z \right )_{\leq n}^{b/2} \leq \left ( 1 - \Delta _z \right )^{b/2}$
we get
$$ \begin{align} \left| (226) \right| &\leq C \int_0^t \left\| \left( 1 - \Delta_z \right)^{b/2} \left< \cdot \right>^a g_s \right\| _{L^2(\mathbb{R}^6)} \nonumber \\ &\quad \times \left\| \left( 1 - \Delta_z \right)^{b/2} \left< z \right>^{a -2 } \left( 2 \left( v - \eta * \boldsymbol{A}_{\alpha_s} \right) \cdot x - \boldsymbol{F}_{f_s, \alpha_s} \cdot v \right) g_s \right\| _{L^2(\mathbb{R}^6)}ds \nonumber \\ &\leq C \int_0^t \left\| g_s \right\| _{W_a^{b,2}(\mathbb{R}^6)} \left\| \left< z \right>^{a -2 } \left( 2 \left( v - \eta * \boldsymbol{A}_{\alpha_s} \right) \cdot x - \boldsymbol{F}_{f_s, \alpha_s} \cdot v \right) g_s \right\| _{W_0^{b,2}(\mathbb{R}^6)}ds. \end{align} $$
By means of (40b) and (46b) we obtain
$$ \begin{align} \left| (226) \right| &\leq C \int_0^t \left( \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) \left\| g_s \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 ds. \end{align} $$
Next, we consider
$$ \begin{align} \Big[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} , T_{f_s, \alpha_s} \Big] &= 2 \left[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} , v \right] \cdot \nabla_x - 2 \left[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} , \eta * \boldsymbol{A}_{\alpha_t} \right] \cdot \nabla_x \nonumber \\ &\quad + 2 \sum_{i=1}^3 \nabla \eta * \boldsymbol{A}_{\alpha_s}^i \left[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} , v^i \right] \cdot \nabla_v \nonumber \\ &\quad + 2 \sum_{i=1}^3 \left[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} , \nabla \eta * \boldsymbol{A}_{\alpha_s}^i \right] v^i \cdot \nabla_v \nonumber \\ &\quad - \left[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} , \left( \nabla K * \widetilde{\rho}_{f_s} + \nabla (\eta * \boldsymbol{A}_{\alpha_s})^2 \right) \right] \cdot \nabla_v. \end{align} $$
Note that
$$ \begin{align} \left\| \left[ v^i , \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \right] f \right\| _{L^2(\mathbb{R}^6)} &\leq \left\| \left( 1 - \Delta_z \right)_{\leq n}^{b-1/2} f \right\| _{L^2(\mathbb{R}^6)} \end{align} $$
because
$\left | \partial _{u_j} \left ( \frac {1 + u^2}{1 + u^2/ n^2} \right )^{b/2} \right | \leq b \left ( \frac {1 + u^2}{1 + u^2/ n^2} \right )^{(b-1)/2}$
. For
$V: \mathbb {R}^3 \rightarrow \mathbb {R}^3$
we, moreover, have
$$ \begin{align} &\left[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} , V(x) \right] \nonumber \\ &\quad = \int_{\mathbb{R}^3} dk \, \mathcal{F}[V](k) e^{-ikx} \left( \frac{\left( 1 + (i \nabla_x + k)^2 - \Delta_v \right)^{b/2}}{\left( 1 + [ (i \nabla_x + k)^2 - \Delta_v ] / n^2 \right)^{b/2}} - \frac{\left( 1 - \Delta_z \right)^{b/2}}{\left( 1 - \Delta_z / n^2 \right)^{b/2} } \right). \end{align} $$
Using the estimate
$$ \begin{align} &\pm \left( \frac{\left( 1 + (i \nabla_x + k)^2 - \Delta_v \right)^{b/2}}{\left( 1 + [ (i \nabla_x + k)^2 - \Delta_v ] / n^2 \right)^{b/2}} - \frac{\left( 1 - \Delta_z \right)^{b/2}}{\left( 1 - \Delta_z / n^2 \right)^{b/2} } \right) \nonumber \\ &\quad = \pm b \int_0^1 dr \, \left( \frac{\left( 1 + (i \nabla_x + r k)^2 - \Delta_v \right)}{\left( 1 + [ (i \nabla_x + r k)^2 - \Delta_v ] / n^2 \right)} \right)^{b/2 - 1} \frac{2 k \cdot (i \nabla_x + r k) (1 - \frac{1}{n^2})}{\left( 1 + [ (i \nabla_x + r k)^2 - \Delta_v ] / n^2 \right)^2} \nonumber \\ &\leq C \left| k \right| \left( \left< k \right>^{b-1} + \left( 1 - \Delta_z \right)^{(b-1/2)} \right) \end{align} $$
we get
$$ \begin{align} \left\| \left[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} , V(x) \right] f \right\| _{L^2(\mathbb{R}^6)} &\leq C \int_{\mathbb{R}^3} \left< k \right>^{b} \left| \mathcal{FT}[V](k) \right| \left\| \left( 1 - \Delta_z \right)^{(b-1)/2} f \right\| _{L^2(\mathbb{R}^6)} dk. \end{align} $$
Since
$$ \begin{align} \int_{\mathbb{R}^3} \left< k \right>^{b} \left| \mathcal{FT}[\nabla \eta * \boldsymbol{A}_{\alpha}](k) \right|dk &\leq C \left\| \alpha \right\| _{\mathfrak{h}_b} \end{align} $$
and
$$ \begin{align} \int_{\mathbb{R}^3} \left< k \right>^b\left| \mathcal{FT}[\nabla K * \widetilde{\rho}_{f}](k) \right| )dk &\leq C \left\| f \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} \end{align} $$
can be obtained by similar bounds as in the proofs of (40b) and (46b) and
$\left ( 1 - \Delta _z \right )_{\leq n}^{b/2} \leq \left ( 1 - \Delta _z \right )^{b/2}$
this leads to
$$ \begin{align} \left\| \Big[ \left( 1 - \Delta_z \right)_{\leq n}^{b/2} , T_{f_s, \alpha_s} \Big] f \right\| _{L^2(\mathbb{R}^6)} &\leq C \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) \left\| \left< v \right> \left( \nabla_x + \nabla_v \right) f \right\| _{W_0^{b-1,2}(\mathbb{R}^6)}. \end{align} $$
Using the Cauchy–Schwarz inequality we obtain
$$ \begin{align} \left| (227) \right| &\leq C \int_0^t \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) \left\| g_s \right\| _{W_a^{b,2}(\mathbb{R}^6)} \nonumber \\ &\quad \times \left\| \left< v \right> \left( \nabla_x + \nabla_v \right) \left< \cdot \right>^a g_s \right\| _{W_0^{b-1,2}(\mathbb{R}^6)} ds. \end{align} $$
Because 
holds pointwise for all
$t \in [0,T]$
and differentiation is a local property we get
$$ \begin{align} \left| (227) \right| &\leq C \left< R_1 \right> \int_0^t \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) \left\| g_s \right\| _{W_a^{b,2}(\mathbb{R}^6)} \nonumber \\ &\quad \times \left\| \left( \nabla_x + \nabla_v \right) \left< \cdot \right>^a g_s \right\| _{W_0^{b-1,2}(\mathbb{R}^6)}ds \nonumber \\ &\leq C \left< R_1 \right> \int_0^t \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) \left\| g_s \right\| ^2_{W_a^{b,2}(\mathbb{R}^6)} ds. \end{align} $$
Collecting the estimates and
$\left ( 1 - \Delta _z \right )_{\leq n}^{b/2} \leq \left ( 1 - \Delta _z \right )^{b/2}$
lead to
$$ \begin{align} &\left\| \left( 1 - \Delta_z \right)_{\leq n}^{b/2} \left< \cdot \right>^a \Phi_t(g) \right\| _{L^2(\mathbb{R}^6)}^2 \nonumber \\ &\quad \leq C \left\| g_0 \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 + C \left< R_1 \right> \int_0^t \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) \left\| g_s \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 ds. \end{align} $$
Note that
$$ \begin{align} \left\| \left( 1 - \Delta_z \right)_{\leq n}^{b/2} f \right\| _{L^2(\mathbb{R}^6)}^2 = \int_0^{\infty} \frac{(1 + \lambda^2)^b}{(1 + \lambda^2/n^2)^b}d \mu_{f}(\lambda) \end{align} $$
by the spectral calculus for
$- \Delta _z$
. Using monotone convergence let us obtain
$$ \begin{align} \lim_{n \rightarrow \infty} \left\| \left( 1 - \Delta_z \right)_{\leq n}^{b/2} f \right\| _{L^2(\mathbb{R}^6)}^2 = \left\| \left( 1 - \Delta_z \right)^{b/2} f \right\| _{L^2(\mathbb{R}^6)}^2 \end{align} $$
and
$$ \begin{align} &\left\| (1 - \Delta_z)^{b/2} \left< \cdot \right>^a \Phi_t(g) \right\| _{L^2(\mathbb{R}^6)}^2 \nonumber \\ &\quad \leq C \left\| g_0 \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 + C \left< R_1 \right> \int_0^t \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) \left\| g_s \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 ds. \end{align} $$
If we choose C in (216) sufficiently large we get
$$ \begin{align} &\left\| (1 - \Delta)^{b/2} \left< \cdot \right>^a \Phi_t(g) \right\| _{L^2(\mathbb{R}^6)}^2 \nonumber \\ &\quad \leq C \left\| g_0 \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 + C \left< R_1 \right> \int_0^t \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) \left\| g_s \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 ds \nonumber \\ &\quad \leq \frac{R_2}{2} + C t \left< R_1 \right> \left( 1 + \left\| f \right\| _{L_t^{\infty} W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha \right\| _{L_t^{\infty} \mathfrak{h}_b}^2 \right) \left\| g \right\| _{L_t^{\infty} W_a^{b,2}(\mathbb{R}^6)}^2 \end{align} $$
and
$$ \begin{align} \left\| \Phi_t(g) \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 &\leq C \left( \left\| g_0 \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 + \left< R_1 \right> \int_0^t \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) \left\| g_s \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 ds \right). \end{align} $$
Together with
$\left \| g \right \| _{L_T^{\infty } W_a^{b,2}(\mathbb {R}^6)}^2 \leq C R_2$
this shows
$\Phi (g) \in L^{\infty } (0,T; W_a^{b,2}(\mathbb {R}^6))$
and
$$ \begin{align} \left\| (1 - \Delta)^{b/2} \left< \cdot \right>^a \Phi_t(g) \right\| _{L_T^{\infty} L^2(\mathbb{R}^6)}^2 \leq R_2 \end{align} $$
for all
$(R_2, T^*)$
satisfying (216) with C chosen sufficiently large.
Contraction property and proof of (213):
Next, we will show that for all
$T^*$
satisfying (216)
$\Phi $
is a contraction on
$(\mathcal {Z}_{T^*,R_1,R_2}, d_{T^*})$
. To this end, consider
$g, g' \in \mathcal {Z}_{T^*,R_1,R_2}$
with
$g_0 - g^{\prime }_{0} = g_0 - g_0 = 0$
and note that
$\Phi $
is a linear mapping. Performing the substitutions
$b \mapsto b-1$
and
$g \mapsto (g - g')$
in (247) then leads to
$$ \begin{align} \left\| \Phi_t(g) - \Phi_t(g') \right\| _{W_a^{b-1,2}(\mathbb{R}^6)}^2 &\leq C \left< R_1 \right> t \left( 1 + \left\| f \right\| _{L_t^{\infty} W_4^{b-3,2}(\mathbb{R}^6)} + \left\| \alpha \right\| _{L_t^{\infty} \mathfrak{h}_{b-1}}^2 \right) d_t(g,g')^2. \end{align} $$
For all
$T^*$
satisfying (216) we consequently get
$$ \begin{align} d_{T^*} \big(\Phi(g), \Phi(g') \big) < d_{T^*} (g,g') , \end{align} $$
proving the contraction property of
$\Phi $
. If we apply the same estimates that led to (247) to the integral equations of (212) and replace
$R_1$
in the estimates by
$R(s)$
as defined in Lemma (6.2) we get
$$ \begin{align} \left\| g_t \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 &\leq C \left\| g_0 \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 + C \int_0^t R(s) \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) \left\| g_s \right\| _{W_a^{b,2}(\mathbb{R}^6)}^2 ds. \end{align} $$
Inequality (213) the follows by Grönwall’s lemma.
Next, we are looking at the linearized equation for the electromagnetic field.
Lemma 6.3. Let
$T>0$
,
$a, b \in \mathbb {N}$
such that
$a \geq 5$
,
$(f, \alpha ) \in C \big ( 0,T; W_a^{b-1,2}(\mathbb {R}^6) \big ) \times C \big ( 0,T; \mathfrak {h}_{b} \cap \dot {\mathfrak {h}}_{-1/2} \big )$
and
$\widetilde {\boldsymbol {J}}_{f_t, \alpha _t}$
be defined as in (16c). The linear equation
$$ \begin{align} i \partial_t \xi_t(k,\lambda) &= \left| k \right| \xi_t(k,\lambda) - \sqrt{ \frac{4 \pi^3}{\left| k \right|}} \mathcal{F}[\eta](k) \boldsymbol{\epsilon}_{\lambda}(k) \cdot \mathcal{F}[\widetilde{\boldsymbol{J}}_{f_t, \alpha_t} ](k) \end{align} $$
with initial datum
$\xi _0 \in \mathfrak {h}_{b} \cap \dot {\mathfrak {h}}_{-1/2}$
has a unique
$C \big ( 0,T; \mathfrak {h}_{b} \cap \dot {\mathfrak {h}}_{-1/2} \big ) \cap C^1 \big ( 0,T; \mathfrak {h}_{b-1} \cap \dot {\mathfrak {h}}_{-1/2} \big )$
solution.
Proof. Note that
$$ \begin{align} &\left\| \widetilde{\boldsymbol{J}}_{f_t, \alpha_t} - \widetilde{\boldsymbol{J}}_{f_s, \alpha_s} \right\| _{W_0^{b-1, 1}(\mathbb{R}^3, \mathbb{C}^3)} \nonumber \\ &\quad \leq C \left( 1 + \left\| \alpha_t \right\| _{\mathfrak{h}_{b-2}} \right) \left\| f_t - f_s \right\| _{W_5^{b-1,2}(\mathbb{R}^6)} + C \left\| f_s \right\| _{W_4^{b-1,2}(\mathbb{R}^6)} \left\| \alpha_t - \alpha_s \right\| _{\mathfrak{h}_{b-2}} \nonumber \\ &\quad \leq C \left( 1 + \left\| \alpha_t \right\| _{\mathfrak{h}_{b-2}} + \left\| f_s \right\| _{W_4^{b-1,2}(\mathbb{R}^6)} \right) \left( \left\| f_t - f_s \right\| _{W_5^{b-1,2}(\mathbb{R}^6)} + \left\| \alpha_t - \alpha_s \right\| _{\mathfrak{h}_{b-2}} \right) \end{align} $$
because of (76c), (40b) and
$\left \| f \right \| _{W_0^{b-1,1}(\mathbb {R}^6)} \leq C \left \| f \right \| _{W_4^{b-1,2}(\mathbb {R}^6)}$
. By means of (28) we have
$$ \begin{align} \left\| \left| \cdot \right|{}^{-1/2} \mathcal{F}[\eta] \, \boldsymbol{\epsilon} \, \mathcal{F}[\widetilde{\boldsymbol{J}}_{f_t, \alpha_t} - \widetilde{\boldsymbol{J}}_{f_s, \alpha_s} ] \right\| _{\mathfrak{h}_b \cap \, \dot{\mathfrak{h}}_{-1/2}} &\leq C \left\| \left< \cdot \right>^{b-1} \mathcal{F}[\widetilde{\boldsymbol{J}}_{f_t, \alpha_t} - \widetilde{\boldsymbol{J}}_{f_s, \alpha_s} ] \right\| _{L^{\infty}(\mathbb{R}^3)} \nonumber \\ &\leq C \left\| \widetilde{\boldsymbol{J}}_{f_t, \alpha_t} - \widetilde{\boldsymbol{J}}_{f_s, \alpha_s} \right\| _{W_0^{b-1, 1}(\mathbb{R}^3, \mathbb{C}^3)}. \end{align} $$
This and the continuity of
$(f, \alpha )$
let us conclude that
$e^{i \left | k \right | s} \sqrt { \frac {4 \pi ^3}{\left | k \right |}} \mathcal {F}[\eta ](k) \boldsymbol {\epsilon }_{\lambda }(k) \mathcal {F}[\widetilde {\boldsymbol {J}}_{f_s, \alpha _s}](k)$
is a
$C \big ( 0,T; \mathfrak {h}_{b} \cap \dot {\mathfrak {h}}_{-1/2} \big )$
function. The Lemma can then be proven similarly as Lemma 6.3.
6.3 Local solutions
Lemma 6.4. Let
$R>0$
and
$a, b \in \mathbb {N}$
such that
$a \geq 5$
and
$b \geq 3$
. For all
$(f_0, \alpha _0) \in W_a^{b,2}(\mathbb {R}^6) \times \mathfrak {h}_b \cap \dot {\mathfrak {h}}_{-1/2}$
such that
$\operatorname {\mathrm {supp}} f_0 \subset A_R$
with
$A_R = \{ (x,v) \in \mathbb {R}^6 , \left | v \right | \leq R \}$
there exists
$T>0$
and a unique
$L^{\infty } \big ( 0,T; W_{a}^{b,2}(\mathbb {R}^6) \big ) \cap C \big ( 0,T; W_{a}^{b-1,2}(\mathbb {R}^6) \big ) \cap C^1 (0,T; W_{a}^{b-2,2}(\mathbb {R}^6)) \times C \big ( 0,T; \mathfrak {h}_{b} \cap \dot {\mathfrak {h}}_{-1/2} \big ) \cap C^1 \big ( 0,T; \mathfrak {h}_{b-1} \cap \dot {\mathfrak {h}}_{-1/2} \big )$
–valued function which satisfies (15) in
$W_a^{b-2,2}(\mathbb {R}^6) \oplus \mathfrak {h}_{b-1} \cap \dot {\mathfrak {h}}_{-1/2}$
with initial datum
$(f_0, \alpha _0)$
.
Proof. Note that
$$ \begin{align} \mathcal{Z}_{T, R^*} &= \Big\{ (f, \alpha) \in C \big( 0,T; W_a^{b-1,2}(\mathbb{R}^6) \big) \times C \big( 0,T; \mathfrak{h}_{b} \cap \dot{\mathfrak{h}}_{-1/2} \big) : (f(t),\alpha(t)) \big|_{t=0} = (f_0, \alpha_0) , \nonumber \\ &\qquad \max \Big\{ \left\| f \right\| _{L_T^{\infty} W_a^{b-1,2}(\mathbb{R}^6)} , \left\| \alpha \right\| _{L_T^{\infty} \mathfrak{h}_b \, \cap \, L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}} \Big\} \leq R^* \Big\} \end{align} $$
with metric
$$ \begin{align} d_T ((f, \alpha), (f', \alpha')) &= \max \left\{ \left\| f - f' \right\| _{L_T^{\infty} W_a^{b-1,2}(\mathbb{R}^6)} , \left\| \alpha - \alpha' \right\| _{L_T^{\infty} \mathfrak{h}_b \, \cap \, L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}} \right\} \end{align} $$
is a Banach space. Next, we consider the solutions of Section 6.2 satisfying
$$ \begin{align} \begin{cases} \partial_t g_t \kern-7pt&= - 2 \left( v - \eta * \boldsymbol{A}_{\alpha_t} \right) \cdot \nabla_x g_t + \boldsymbol{F}_{f_t, \alpha_t} \cdot \nabla_v g_t , \\ i \partial_t \xi_t(k,\lambda) \kern-7pt&= \left| k \right| \xi_t(k,\lambda) - \sqrt{ \frac{4 \pi^3}{\left| k \right|}} \mathcal{F}[\eta](k) \boldsymbol{\epsilon}_{\lambda}(k) \cdot \mathcal{F}[\widetilde{\boldsymbol{J}}_{f_t, \alpha_t} ](k) \end{cases} \end{align} $$
with initial datum
$(f_0, \alpha _0)$
and define the mapping
$\Phi : \mathcal {Z}_{R^*,T} \rightarrow L^{\infty } \big ( 0,T; W_{a}^{b,2}(\mathbb {R}^6) \big ) \cap C \big ( 0,T; W_{a}^{b-1,2}(\mathbb {R}^6) \big ) \cap C^1 (0,T; W_{a}^{b-2,2}(\mathbb {R}^6)) \times C \big ( 0,T; \mathfrak {h}_{b} \cap \dot {\mathfrak {h}}_{-1/2} \big ) \cap C^1 \big ( 0,T; \mathfrak {h}_{b-1} \cap \dot {\mathfrak {h}}_{-1/2} \big )$
by
$$ \begin{align} \Phi \begin{pmatrix} f \\ \alpha \end{pmatrix} &= \begin{pmatrix} g \\ \xi \end{pmatrix}. \end{align} $$
Below, we will show that for
$$ \begin{align} R^* \geq C \left( \left\| f_0 \right\| _{W_a^{b-1,2}} + \left\| \alpha_0 \right\| _{\mathfrak{h}_b \, \cap \,\dot{\mathfrak{h}}_{-1/2} } \right) \quad \text{and} \quad T \leq \exp \left( - C \left( \left< R \right> + \left< R^* \right>^2 + \left\| f_0 \right\| _{W_a^{b,2}(\mathbb{R}^6)} \right) \right) \end{align} $$
with
$C>0$
chosen sufficiently large the mapping
$\Phi $
is a contraction on
$\mathcal {Z}_{T, R^*}$
. This leads to a unique fixed point satisfying
$d_T ((f, \alpha ), (g, \xi )) = 0$
. Replacing
$(g, \xi )$
by
$(f, \alpha )$
in the integral version of (257) and then proves that
$(f, \alpha )$
satisfies (15).
Well-definedness of
$\Phi $
:
By means of (213) and
$(f,\alpha ) \in \mathcal {Z}_{T,R^*}$
we get
$$ \begin{align} \left\| g_t \right\| _{W_a^{b-1,2}(\mathbb{R}^6)} &\leq C \left\| f_0 \right\| _{W_a^{b-1,2}(\mathbb{R}^6)} e^{C t \left< R^* \right>^2 \left< R \right> e^{C t \left< R^* \right>^2}}. \end{align} $$
Using Duhamel’s formula, (28) and (76c) we get by similar estimates as in the proof of Lemma (6.3)
$$ \begin{align} \left\| \xi_t \right\| _{\mathfrak{h}_{b} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} &\leq C \left\| \alpha_0 \right\| _{\mathfrak{h}_{b} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C \int_0^t \left\| \widetilde{\boldsymbol{J}}_{f_s, \alpha_s} \right\| _{W_0^{b-1, 1}(\mathbb{R}^3, \mathbb{C}^3)}ds \nonumber \\ &\leq C \left\| \alpha_0 \right\| _{\mathfrak{h}_{b} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C t \left( 1 + \left\| \alpha \right\| _{L_t^{\infty} \mathfrak{h}_{b -2}} \right) \left\| f \right\| _{L_t^{\infty} W_5^{b-1,2}(\mathbb{R}^6)} \nonumber \\ &\leq C \left\| \alpha_0 \right\| _{\mathfrak{h}_{b} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C t \left< R^* \right>^2. \end{align} $$
For C in (259) chosen sufficiently large this shows
$\left \| g \right \| _{L_T^{\infty } W_a^{b-1,2}(\mathbb {R}^6)} \leq R^*$
and
$\left \| \xi \right \| _{L_T^{\infty } \mathfrak {h}_b \, \cap \, L_T^{\infty } \dot {\mathfrak {h}}_{-1/2}} \leq R^*$
, proving that
$\Phi $
maps
$\mathcal {Z}_{T,R^*}$
into itself.
Contraction property of
$\Phi $
:
Let
$(f, \alpha )$
,
$(f', \alpha ') \in \mathcal {Z}_{T, R^*}$
and denote their images under the mapping
$\Phi $
by
$(g, \xi )$
and
$(g', \xi ')$
. In the following, we prove
$d_T ((g, \xi ), (g', \xi ')) < d_T ((f, \alpha ), (f', \alpha '))$
for all
$T \geq 0$
satisfying (259) with sufficiently large constant. Using Duhamel’s formula, (28) and (253) we get
$$ \begin{align} &\left\| \xi - \xi' \right\| _{L_T^{\infty} \mathfrak{h}_{b} \, \cap \, L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}} \nonumber \\ &\quad \leq C \sup_{t \in [0,T]} \int_0^t \left\| \left| \cdot \right|{}^{-1} \left< \cdot \right>^{3/2} \mathcal{F}[\eta] \right\| _{L^2(\mathbb{R}^3)} \left\| \left< \cdot \right>^{b-1} \mathcal{F}[\widetilde{\boldsymbol{J}}_{f_s, \alpha_s} - \widetilde{\boldsymbol{J}}_{f_s', \alpha_s'}] \right\| _{L^{\infty}(\mathbb{R}^3, \mathbb{C}^3)}ds \nonumber \\ &\quad \leq C \sup_{t \in [0,T]} \int_0^t \left\| \widetilde{\boldsymbol{J}}_{f_s, \alpha_s} - \widetilde{\boldsymbol{J}}_{f_s', \alpha_s'} \right\| _{W_0^{b-1, 1}(\mathbb{R}^3, \mathbb{C}^3)}ds \nonumber \\ &\quad \leq C T \left( 1 + \left\| \alpha \right\| _{L_T^{\infty} \mathfrak{h}_{b-2}} + \left\| f' \right\| _{L_T^{\infty} W_4^{b-1,2}(\mathbb{R}^6)} \right) d_T ((f, \alpha), (f', \alpha')) \nonumber \\ &\quad \leq C T \left< R^* \right> \left( \left\| f - f' \right\| _{L_T^{\infty} W_5^{b-1,2}(\mathbb{R}^6)} + \left\| \alpha - \alpha' \right\| _{L_T^{\infty} \mathfrak{h}_{b-2}} \right) \nonumber \\ &\quad < d_T ((f, \alpha), (f', \alpha')). \end{align} $$
By means of
$$ \begin{align} D_z^{\sigma} \left( g_t - g^{\prime}_t \right) &= - \int_0^t T_{f_s' , \alpha_s'}[D_z^{\sigma} (g_s - g_s' )]\,ds - \int_0^t \big[ D_z^{\sigma} , T_{f_s' , \alpha_s'} \big] [g_s - g_s']\,ds \nonumber \\ &\quad - \int_0^t D_z^{\sigma} \big( T_{f_s , \alpha_s} - T_{f_s' , \alpha_s'} \big) [g_s]\,ds \end{align} $$
with
$T_{f,\alpha }$
being defined as in (221) we get for
$0 \leq \left | \sigma \right | \leq b-1$
and
$t \in [0,T]$
$$ \begin{align} &\left\| D_z^{\sigma} (g_t - g^{\prime}_t) \right\| _{W_a^{0,2}}^2 \nonumber \\ &\quad = - 2 \int_0^t \int_{\mathbb{R}^6} \left< z \right>^{2a} (D_z^{\sigma} (g_s - g^{\prime}_s))(x,v) \big( T_{f_s' , \alpha_s'}[D_z^{\sigma} (g_s - g_s' )] \big)(x,v)\,dx\,dv\,ds \end{align} $$
$$ \begin{align} &\qquad - 2 \int_0^t \int_{\mathbb{R}^6} \left< z \right>^{2a} (D_z^{\sigma} (g_s - g^{\prime}_s))(x,v) \big( \big[ D_z^{\sigma} , T_{f_s' , \alpha_s'} \big] [g_s - g_s'] \big)(x,v)\,dx\,dv\,ds \end{align} $$
$$ \begin{align} &\qquad - 2 \int_0^t \int_{\mathbb{R}^6} \left< z \right>^{2a} (D_z^{\sigma} (g_s - g^{\prime}_s))(x,v) \big( D_z^{\sigma} \big( T_{f_s , \alpha_s} - T_{f_s' , \alpha_s'} \big) [g_s] \big)(x,v)\,dx\,dv\,ds. \end{align} $$
Using (224), (228), (40b) and (46b) we estimate
$$ \begin{align} \left| (\text{264a}) \right| &= \left| \int_0^t \int_{\mathbb{R}^6} T_{f^{\prime}_s, \alpha^{\prime}_s}\big[ \left< z \right>^{2a} \big] \left| (D_z^{\sigma} (g_s - g^{\prime}_s))(x,v) \right|^2 \right|dx\,dv\,ds \nonumber \\ &\leq C \int_0^t \left( 1 + \left\| f^{\prime}_s \right\| _{W_4^{0,2}(\mathbb{R}^6)} + \left\| \alpha^{\prime}_s \right\| _{\mathfrak{h}}^2 \right) \int_{\mathbb{R}^6} \left< z \right>^{2a} \left| (D_z^{\sigma} (g_s - g^{\prime}_s))(x,v) \right|^2 dx\,dv\,ds \nonumber \\ &\leq C t \left( 1 + \left\| f' \right\| _{L_t^{\infty} W_4^{0,2}(\mathbb{R}^6)} + \left\| \alpha' \right\| _{L_t^{\infty} \mathfrak{h}}^2 \right) \left\| g - g' \right\| _{ L_t^{\infty} W_a^{b-1,2}(\mathbb{R}^6)}^2 \nonumber \\ &\leq C t \left< R^* \right>^2 \left\| g - g' \right\| _{ L_t^{\infty} W_a^{b-1,2}(\mathbb{R}^6)}^2. \end{align} $$
Due to (40b) and (46b) and the chain rule of differentiation we have
$$ \begin{align} &\left\| \left< v \right> ^{-1} \left< z \right> ^a \big[ D_z^{\sigma} , T_{f_s' , \alpha_s'} \big] [g_s - g_s'] \right\| _{L^2(\mathbb{R}^6)} \nonumber \\ &\quad \leq C \left( 1 + \left\| f^{\prime}_s \right\| _{W_4^{b-3,2}(\mathbb{R}^6)} + \left\| \alpha^{\prime}_s \right\| _{\mathfrak{h}_{b-1}}^2 \right) \left\| g_s - g^{\prime}_s \right\| _{ W_a^{b-1,2}(\mathbb{R}^6)} \nonumber \\ &\quad \leq C \left< R^* \right>^2 \left\| g_s - g^{\prime}_s \right\| _{ W_a^{b-1,2}(\mathbb{R}^6)}. \end{align} $$
Note that
$$ \begin{align} \operatorname{\mathrm{supp}} (g_s - g^{\prime}_s) \subset A_{R(s)} \quad \text{with} \quad R(s) &\leq C \left< R \right> e^{C s \left< R^* \right>^2} \end{align} $$
because of Lemma 6.2 and
$(f,\alpha ), (f',\alpha ') \in \mathcal {Z}_{T,R^*}$
. Since differentiation with respect to
$D_z^{\sigma }$
is a local operation we get
$$ \begin{align} \left\| \left< v \right> \left< z \right> ^a D_z^{\sigma} (g_s - g_s') \right\| _{L^2(\mathbb{R}^6)} &\leq R(s) \left\| \left< z \right> ^a D_z^{\sigma} (g_s - g_s') \right\| _{L^2(\mathbb{R}^6)}. \end{align} $$
Together with the Cauchy–Schwarz inequality this leads to
$$ \begin{align} \left| (264b) \right| &\leq C t \left< R \right> \left< R^* \right>^2 e^{C t \left< R^* \right>^2} \left\| g - g' \right\| _{ L_t^{\infty} W_a^{b-1,2}(\mathbb{R}^6)}^2. \end{align} $$
Recall that
$$ \begin{align} T_{f, \alpha} - T_{f', \alpha'} &= - 2 \eta * \boldsymbol{A}_{\alpha - \alpha'} \cdot \nabla_x - \big( \boldsymbol{F}_{f, \alpha} - \boldsymbol{F}_{f' , \alpha'} \big) \cdot \nabla_v \end{align} $$
and
$$ \begin{align} \boldsymbol{F}_{f, \alpha} - \boldsymbol{F}_{f' , \alpha'} &= \nabla K * \widetilde{\rho}_{f - f'} (x) - 2 \sum_{i=1}^3 ( \nabla \eta * \boldsymbol{A}^i_{\alpha - \alpha'} ) (x) \left( v^i - \eta * \boldsymbol{A}^i_{\alpha}(x) \right) \nonumber \\ &\quad + 2 \sum_{i=1}^3 ( \nabla \eta * \boldsymbol{A}^i_{\alpha'} ) (x) \eta * \boldsymbol{A}^i_{\alpha - \alpha'}(x). \end{align} $$
Using (40b) and (46b) and the chain rule we get
$$ \begin{align} \left| \big( D_z^{\sigma} \big( T_{f_s , \alpha_s} - T_{f_s' , \alpha_s'} \big) [g_s] \big) (x,v) \right| &\leq C \left< v \right> \left( \left\| \alpha_s - \alpha^{\prime}_s \right\| _{\mathfrak{h}_{b-1}} + \left\| f_s - f^{\prime}_s \right\| _{W_a^{b-3,2}} \right) \nonumber \\ &\quad \times \left( 1 + \left\| \alpha_s \right\| _{\mathfrak{h}_{b-1}} + \left\| \alpha^{\prime}_s \right\| _{\mathfrak{h}_{b-1}} \right) \sum_{\left| \nu \right| \leq b} \left| D_z^{\nu} g_s(x,v) \right| \end{align} $$
and
$$ \begin{align} &\left\| \left< v \right>^{-1} \left< z \right>^a D_z^{\sigma} \big( T_{f_s , \alpha_s} - T_{f_s' , \alpha_s'} \big) [ g_s] \right\| _{L^2(\mathbb{R}^6)} \nonumber \\ &\quad \leq C \left( 1 + \left\| \alpha_s \right\| _{\mathfrak{h}_{b-1}} + \left\| \alpha^{\prime}_s \right\| _{\mathfrak{h}_{b-1}} \right) \left\| g_t \right\| _{W_a^{b,2}(\mathbb{R}^6)} \left( \left\| \alpha_s - \alpha^{\prime}_s \right\| _{\mathfrak{h}_{b-1}} + \left\| f_s - f^{\prime}_s \right\| _{W_a^{b-3,2}(\mathbb{R}^6)} \right). \end{align} $$
By means of the Cauchy–Schwarz inequality and (268) we obtain
$$ \begin{align} \left| (\text{264c}) \right| &\leq 2 \int_0^t \left\| \left< v \right> \left< z \right> ^a D_z^{\sigma} (g_s - g_s') \right\| _{L^2(\mathbb{R}^6)} \left\| \left< v \right>^{-1} \left< z \right>^a D_z^{\sigma} \big( T_{f_s , \alpha_s} - T_{f_s' , \alpha_s'} \big) [g_s] \right\| _{L^2(\mathbb{R}^6)} ds \nonumber \\ &\leq C t \, R(t) \left( 1 + \left\| \alpha \right\| ^2_{L_t^{\infty} \mathfrak{h}_{b-1}} + \left\| \alpha' \right\| ^2_{L_t^{\infty} \mathfrak{h}_{b-1}} \right) \left\| g \right\| _{L_t^{\infty} W_a^{b,2}(\mathbb{R}^6)} \nonumber \\ &\quad \times \left\| g - g' \right\| _{ L_t^{\infty} W_a^{b-1,2}(\mathbb{R}^6)} \left( \left\| \alpha - \alpha' \right\| _{L_t^{\infty} \mathfrak{h}_{b-1}} + \left\| f - f' \right\| _{L_t^{\infty} W_a^{b-3,2}(\mathbb{R}^6)} \right). \end{align} $$
In combination with (213) this leads to
$$ \begin{align} \left| (\text{264c}) \right| &\leq C t \, R(t) \left( 1 + \left\| \alpha \right\| ^2_{L_t^{\infty} \mathfrak{h}_{b-1}} + \left\| \alpha' \right\| ^2_{L_t^{\infty} \mathfrak{h}_{b-1}} \right) \left\| g_0 \right\| _{W_a^{b,2}} \nonumber \\ &\quad \times e^{C \int_0^t R(s) \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right)ds } \left\| g - g' \right\| _{ L_t^{\infty} W_a^{b-1,2}(\mathbb{R}^6)} d_t((f,\alpha), (f', \alpha')) \nonumber \\ &\leq C t \left< R^* \right>^2 e^{2 \left< R \right> e^{C t \left< R^* \right>^2}} \left\| g_0 \right\| _{W_a^{b,2}} \left\| g - g' \right\| _{ L_t^{\infty} W_a^{b-1,2}(\mathbb{R}^6)} d_t((f,\alpha), (f', \alpha')). \end{align} $$
Collecting the estimates, taking the sum
$\sum _{\left | \sigma \right | \leq b-1}$
and dividing by
$\left \| g - g' \right \| _{ L_t^{\infty } W_a^{b-1,2}(\mathbb {R}^6)}$
gives
$$ \begin{align} \left\| g - g' \right\| _{ L_t^{\infty} W_a^{b-1,2}(\mathbb{R}^6)} &\leq C t \left< R \right> \left< R^* \right>^2 e^{C t \left< R^* \right>^2} \left\| g - g' \right\| _{ L_t^{\infty} W_a^{b-1,2}(\mathbb{R}^6)} \nonumber \\ &\quad + C t \left< R^* \right>^2 e^{2 \left< R \right> e^{C t \left< R^* \right>^2}} \left\| g_0 \right\| _{W_a^{b,2}} d_t((f,\alpha), (f', \alpha')) \nonumber \\ &\leq t e^{C \left( \left< R \right> + \left< R^* \right>^2 \right) e^{C t \left< R^* \right>^2}} \left( 1 + \left\| g_0 \right\| _{W_a^{b,2}} \right) \nonumber \\ &\quad \times \left( \left\| g - g' \right\| _{ L_t^{\infty} W_a^{b-1,2}(\mathbb{R}^6)} + d_t((f,\alpha), (f', \alpha')) \right). \end{align} $$
For
$T \geq 0$
satisfying (259) with
$C> 0$
chosen large enough we then obtain
$$ \begin{align} \left\| g - g' \right\| _{L_T^{\infty} W_a^{b-1,2}} &\leq \frac{1}{2} \left\| g - g' \right\| _{L_T^{\infty} W_a^{b-1,2}} + \frac{1}{4} d_T((f, \alpha), (f', \alpha')) \end{align} $$
and therefore
$\left \| g - g' \right \| _{L_T^{\infty } W_a^{b-1,2}} < d_T((f, \alpha ), (f', \alpha '))$
.
6.4 Global solutions
Proof of Proposition 2.2.
The existence of a unique local solution is ensured by Lemma 6.4. By similar means as in the proof of [Reference Cazenave and Haraux15, Theorem 4.3.4] it is straightforward to prove that
$T_{\text {max}}$
is either infinite or
$\lim _{t \nearrow T_{\text {max}}} \left ( \left \| f_t \right \| _{W_a^{b,2}} + \left \| \alpha _t \right \| _{\mathfrak {h}_b \, \cap \,\dot {\mathfrak {h}}_{-1/2} } \right ) = \infty $
. In the following we assume that the local solution of Lemma 6.4 exists until time T and show
$\left \| f_t \right \| _{W_a^{b,2}} + \left \| \alpha _t \right \| _{\mathfrak {h}_b \, \cap \,\dot {\mathfrak {h}}_{-1/2} } < + \infty $
for all
$t \in [0,T]$
. On the interval of existence the conservation of the energy follows from a straightforward calculation. If we define the characteristics
$(X_t(x,v), V_t(x,v))$
by (208) with
$(f_t, \alpha _t)$
satisfying (15) and initial datum
$(X_t(x,v), V_t(x,v)) \big |_{t=0} = (x,v)$
we can write the particle distribution in terms of the corresponding backward characteristics as
$f_t(x,v) = f_0 \big ( X_t^{-1}(x,v) , V_t^{-1}(x,v)) \big )$
. The conservation of the
$L^p$
–norms then follows from a change of coordinates and the fact that the flow is measure preserving. Next, we will show the finiteness of the
$W_a^{b,2}(\mathbb {R}^6)$
and
$\mathfrak {h}_b \, \cap \,\dot {\mathfrak {h}}_{-1/2}$
norms. By means of (208) and (76a) and the conservation of mass and energy we get
$$ \begin{align} \left< V_t(x,v) \right> &\leq C \left< v \right> + C \left( 1 + \mathcal{E}^{\mathrm{{VM}}}[f_0, \alpha_0] + C \left\| f \right\| _{L^1(\mathbb{R}^6)}^2 \right) \int_0^{t} \left< V_s(x,v) \right> ds, \end{align} $$
leading to
$$ \begin{align} \left< V_t(x,v) \right> &\leq C \left<v \right>e^{C \left( 1 + \mathcal{E}^{\mathrm{{VM}}}[f_0 , \alpha_0] + C \left\| f_0 \right\| _{L^1(\mathbb{R}^6)}^2 \right) t} \end{align} $$
by Grönwall’s lemma. The first equation of (208), (40a), (75a), the conservation of mass and energy as well as (279) let us obtain
$$ \begin{align} \left| X_t(x,v) \right| &\leq \left| x \right| + C \left( 1 + \mathcal{E}^{\mathrm{{VM}}}[f_0, \alpha_0] + C \left\| f_0 \right\| _{L^1(\mathbb{R}^6)}^2 \right) \left( \left| t \right| + \int_0^{t} \left| V_s(x,v) \right| ds\right) \nonumber \\ &\leq \left< (x,v) \right> e^{C \left( 1 + \mathcal{E}^{\mathrm{{VM}}}[f_0, \alpha_0] + C \left\| f_0 \right\| _{L^1(\mathbb{R}^6)}^2 \right) \left< t \right>}. \end{align} $$
If we use the previous estimates and the fact that the flow is measure preserving we get
$$ \begin{align} \left\| f_t \right\| _{W_a^{0,2}(\mathbb{R}^6)}^2 &= \int_{\mathbb{R}^6} \left< (x,v) \right>^{2a} \left| f_0 \big( X_t^{-1}(x,v) , V_t^{-1}(x,v)) \big) \right|^2 dx\,dv \nonumber \\ &= \int_{\mathbb{R}^6} \left< \big( X_t(x,v) , V_t(x,v) \big) \right>^{2a} \left| f_0 (x,v) \right|^2 dx\,dv \nonumber \\ &\leq e^{C \left( 1 + \mathcal{E}^{\mathrm{{VM}}}[f_0, \alpha_0] + C \left\| f_0 \right\| _{L^1(\mathbb{R}^6)}^2 \right) \left< t \right>} \left\| f_0 \right\| _{W_a^{0,2}(\mathbb{R}^6)}^2 \end{align} $$
by a change of coordinates. Due to (15), Duhamel’s formula and (28) we have
$$ \begin{align} \left\| \alpha_t \right\| _{\mathfrak{h}_{b} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} &\leq \left\| \alpha_0 \right\| _{\mathfrak{h}_{b} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C \int_0^t \left\| \widetilde{\boldsymbol{J}}_{f_s, \alpha_s} \right\| _{W_0^{b-1, 1}(\mathbb{R}^3, \mathbb{C}^3)} ds. \end{align} $$
By (76b), (76c) and the conservation of the
$L^p$
–norms and energy we obtain
$$ \begin{align} \left\| \alpha_t \right\| _{\mathfrak{h}_1 \, \cap \, \dot{\mathfrak{h}}_{-1/2}} &\leq \left\| \alpha_0 \right\| _{\mathfrak{h}_1 \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C t \left( 1 + \mathcal{E}^{\mathrm{{VM}}}[f_0, \alpha_0] + C \left\| f_0 \right\| _{L^1(\mathbb{R}^6)}^2 \right) \quad \text{for} \; b = 1 \end{align} $$
and
$$ \begin{align} \left\| \alpha_t \right\| _{\mathfrak{h}_{b} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} &\leq \left\| \alpha_0 \right\| _{\mathfrak{h}_{b} \, \cap \, \dot{\mathfrak{h}}_{-1/2}} + C t \left( 1 + \left\| \alpha \right\| _{L_t^{\infty}\mathfrak{h}_{b-2}} \right) \left\| f \right\| _{L_t^{\infty} W_5^{b-1,2}(\mathbb{R}^6)} \quad \text{for} \; b \geq 2. \end{align} $$
We consequently have
$$ \begin{align} \left\| f \right\| _{L_T^{\infty} W_a^{0,2}(\mathbb{R}^6)} + \left\| \alpha \right\| _{L_T^{\infty} \mathfrak{h}_1 \, \cap \, L_T^{\infty} \dot{\mathfrak{h}}_{-1/2}} < + \infty. \end{align} $$
The second inequality of (284) and (see (213))
$$ \begin{align} \left\| f_t \right\| _{W_a^{b,2}(\mathbb{R}^6)} &\leq C \left\| f_0 \right\| _{W_a^{b,2}} e^{C \int_0^t R(s) \left( 1 + \left\| f_s \right\| _{W_4^{b-2,2}(\mathbb{R}^6)} + \left\| \alpha_s \right\| _{\mathfrak{h}_b}^2 \right) ds} \end{align} $$
with
$R(t) \leq C \left < R \right> e^{C \int _0^t \left ( \left \| f_s \right \| _{W_4^{0,2}(\mathbb {R}^6)} + \left \| \alpha _s \right \| _{ \mathfrak {h}}^2 \right )ds}$
let us then iteratively increase the values of
$\beta $
until we get
$\sup _{t \in [0,T]} \left \{ \left \| f_t \right \| _{W_a^{b,2}} + \left \| \alpha _t \right \| _{\mathfrak {h}_b \, \cap \,\dot {\mathfrak {h}}_{-1/2} } \right \} < + \infty $
.
A Equivalent forms of the Vlasov–Maxwell equations
Let
$(f, \alpha )$
satisfy (15) with
$\eta (x) = - \delta (x)$
. We define
$$ \begin{align} \boldsymbol{E}^{\perp}_{\alpha_t}(x) &= \frac{i}{(2 \pi)^{3/2}} \sum_{\lambda = 1,2} \int \sqrt{\frac{\left| k \right|}{2}} \boldsymbol{\epsilon}_{\lambda}(k) \left( e^{i k x} \alpha(k,\lambda) - e^{- i k x} \overline{\alpha(k, \lambda)} \right)dk \end{align} $$
and
$g_t(x,v) = f_t(x, v - \boldsymbol {A}_{\alpha _t}(x) )$
satisfying
$\widetilde {\rho }_{f_t} = \widetilde {\rho }_{g_t}$
and
$\widetilde {\boldsymbol {J}}_{f_t, \alpha _t}(x) = 2 \int _{\mathbb {R}^3} v g_t(x,v)dv$
. Using
$$ \begin{align} &\boldsymbol{F}_{f_t, \alpha_t}(x, v - \boldsymbol{A}_{\alpha_t}(x)) - 2 \sum_{j=1}^3 v^j \nabla^j \boldsymbol{A}_{\alpha_t}(x) \nonumber \\ &\quad = \frac{1}{4 \pi} \nabla | \cdot |^{-1} * \widetilde{\rho}_{f_t}(x) + 2 \sum_{j=1}^3 \left( v^j \nabla \boldsymbol{A}^j_{\alpha_t}(x) - v^j \nabla^j \boldsymbol{A}_{\alpha_t}(x) \right) \nonumber \\ &\quad = \frac{1}{4 \pi} \nabla | \cdot |^{-1} * \widetilde{\rho}_{f_t}(x) + 2 v \times \left( \nabla \times \boldsymbol{A}_{\alpha_t} \right)(x) \end{align} $$
and (15) we obtain
$$ \begin{align} \begin{cases} \partial_t g_t \kern-7pt&= - 2 v \cdot \nabla_x g_t + \left( \boldsymbol{E}^{\perp}_{\alpha_t} + \frac{1}{4 \pi} \nabla | \cdot |^{-1} * \widetilde{\rho}_{g_t} + 2 v \times \left( \nabla \times \boldsymbol{A}_{\alpha_t} \right) (x) \right) \cdot \nabla_v g_t \\ \partial_t \boldsymbol{A}_{\alpha_t} \kern-7pt&= - \boldsymbol{E}^{\perp}_{\alpha_t} \\ \partial_t \boldsymbol{E}^{\perp}_{\alpha_t} \kern-7pt&= - \Delta \boldsymbol{A}_{\alpha_t} + 2 \left( 1 - \nabla \text{div} \Delta^{-1} \right) \int_{\mathbb{R}^3} v g_t(\cdot,v) dv \end{cases}. \end{align} $$
Next we define
$h_t(x,v) = g_t(x, v/2) / (32 \pi )$
. Note that
$$ \begin{align} 2 \int_{\mathbb{R}^3} v g_t(\cdot,v) dv = \frac{1}{8} \int_{\mathbb{R}^3} v g_t(\cdot,v/2) dv = 4 \pi \int_{\mathbb{R}^3} v h_t(\cdot,v) dv \end{align} $$
and
$\widetilde {\rho }_{g_t} = 4 \pi \widetilde {\rho }_{h_t}$
. Together with
$(32 \pi ) \left ( v h_t \right )(x,v) = \left ( 2 v g_t\right )(x,v/2) /(32 \pi )$
and
$(32 \pi ) \left ( \nabla _v h_t \right )(x,v) = \frac {1}{2} (\nabla _v g)(x,v/2)$
we get
$$ \begin{align} \begin{cases} \partial_t h_t \kern-7pt&= - v \cdot \nabla_x h_t + 2 \left( \boldsymbol{E}^{\perp}_{\alpha_t} + \nabla | \cdot |^{-1} * \widetilde{\rho}_{h_t} + v \times \left( \nabla \times \boldsymbol{A}_{\alpha_t} \right) (x) \right) \cdot \nabla_v h_t \\ \partial_t \boldsymbol{A}_{\alpha_t} \kern-7pt&= - \boldsymbol{E}^{\perp}_{\alpha_t} \\ \partial_t \boldsymbol{E}^{\perp}_{\alpha_t} \kern-7pt&= - \Delta \boldsymbol{A}_{\alpha_t} + \left( 1 - \nabla \text{div} \Delta^{-1} \right) 4 \pi \int_{\mathbb{R}^3} v h_t(\cdot,v) dv \end{cases}. \end{align} $$
If we define the electric and magnetic field in the usual way by
$$ \begin{align} \boldsymbol{E}_t = \boldsymbol{E}^{\perp}_{\alpha_t} + \nabla | \cdot |^{-1} * \widetilde{\rho}_{h_t} \quad \text{and} \quad \boldsymbol{B}_t = \nabla \times \boldsymbol{A}_{\alpha_t} \end{align} $$
we have that the equations
$$ \begin{align} \nabla \cdot \boldsymbol{B}_t = 0 \quad \text{and} \quad \partial_t \boldsymbol{B}_t + \nabla \times \boldsymbol{E}_t = 0 \end{align} $$
are automatically satisfied. Using that we are working in the Coulomb gauge and
$- \Delta \frac {1}{4 \pi \left | x- y \right |} = \delta (x-y)$
let us obtain
$$ \begin{align} \nabla \cdot \boldsymbol{E}_t = - 4 \pi \widetilde{\rho}_{h_t}. \end{align} $$
Since
$\nabla \times \boldsymbol {B}_t = \nabla \text {div} \boldsymbol {A}_{\alpha _t} - \Delta \boldsymbol {A}_{\alpha _t} = - \Delta \boldsymbol {A}_{\alpha _t} $
and
$\partial _t \nabla | \cdot |^{-1} * \widetilde {\rho }_{h_t} = \nabla \text {div} \Delta ^{-1} 4 \pi \int v h_t(\cdot ,v)dv$
we, moreover, get
$$ \begin{align} \partial_t \boldsymbol{E}_{t} - \nabla \times \boldsymbol{B}_t &= 4 \pi \int v h_t(\cdot,v)dv. \end{align} $$
In total, this shows that
$(h, \boldsymbol {E}, \boldsymbol {B})$
satisfies (1) with
$c= 1$
,
$q= -1$
and
$m=1/2$
.
Likewise, (15) with
$\eta (x) = - \delta (x)$
can be obtained from (1) with
$c= 1$
,
$q= -1$
and
$m=1/2$
by performing the inverse of the coordinate transformation on the phase space density described above, rewriting Maxwell’s equations in terms of the transverse electric field
$\boldsymbol {E}_t^{\perp }$
and the vector potential
$\boldsymbol {A}_t$
in the Coulomb gauge, and then defining
$$ \begin{align} \alpha_t(k,\lambda) &= \frac{1}{\sqrt{2}} \boldsymbol{\epsilon}_{\lambda}(k) \cdot \left( \left| k \right|{}^{1/2} \mathcal{F}[\boldsymbol{A}_t](k) - i \left| k \right|{}^{-1/2} \mathcal{F}[\boldsymbol{E}_t^{\perp}](k) \right). \end{align} $$
Further details regarding this representation of Maxwell’s equations can be found in [Reference Cohen-Tannoudji, Dupont-Roc and Grynberg20, Chapter I.C.].
B Auxiliary estimates
Lemma B.1. Let
$N \in \mathbb {N}$
,
$\varepsilon = N^{-1/3}$
and
${\mathsf {D}} = \big ( 1 - \varepsilon ^2 \Delta \big )^{1/2}$
. Let
$\beta \in \mathbb {N}_0^3$
,
$\omega _N$
regular enough and
$W_N$
denote its Wigner transform. Then,
$$ \begin{align} \sup \left\{ \left\| {\mathsf{D}} \omega_N {\mathsf{D}} \right\| _{\mathfrak{S}^1} , \left\| {\mathsf{D}} \omega_N i \varepsilon \nabla \right\| _{\mathfrak{S}^1}, \left\| {\mathsf{D}} \omega_N \right\| _{\mathfrak{S}^1} \right\} &\leq C N \sum_{j=0}^6 \varepsilon^j \left\| W_N \right\| _{H_6^j} , \end{align} $$
$$ \begin{align} \left\| \left[ \hat{x} , {\mathsf{D}} \omega_N {\mathsf{D}} \right] \right\| _{\mathfrak{S}^1} &\leq C N \sum_{j=1}^7 \varepsilon^j \left\| W_N \right\| _{H_6^{j}} , \end{align} $$
$$ \begin{align} \left\| \left[ i \nabla , {\mathsf{D}} \omega_N {\mathsf{D}} \right] \right\| _{\mathfrak{S}^1} &\leq C N \sum_{j=0}^6 \varepsilon^j \left\| W_N \right\| _{H_6^{j+1}} , \end{align} $$
$$ \begin{align} \left\| \left[ \hat{x} , {\mathsf{D}} \omega_N {\mathsf{D}} i \nabla \right] \right\| _{\mathfrak{S}^1} &\leq C N \sum_{j=0}^7 \varepsilon^j \left\| W_N \right\| _{H_7^{j+1}} , \end{align} $$
$$ \begin{align} \varepsilon^{\left| \beta \right|} \left\| \left( 1 + x^2 \right) \nabla^{\beta} \left[ \hat{x} , \left[ \hat{x} , \omega_N \right] \right] \right\| _{\mathfrak{S}^{2}} &\leq C N^{1/2} \sum_{j=2}^{4 + \left| \beta \right|} \varepsilon^j \left\| W_N \right\| _{H_{2 + \left| \beta \right|}^{j}} , \end{align} $$
$$ \begin{align} \varepsilon^{\left| \beta \right|} \left\| \left( 1 + x^2 \right) \nabla^{\beta} \left[ \hat{x} , \left[ \hat{x} , \omega_N \right] \right] i \varepsilon \nabla \right\| _{\mathfrak{S}^{2}} &\leq C N^{1/2} \sum_{j=2}^{5 + \left| \beta \right|} \varepsilon^j \left\| W_N \right\| _{H_{3 + \left| \beta \right|}^{j}} , \end{align} $$
$$ \begin{align} \varepsilon^{\left| \beta \right|} \left\| \left( 1 + x^2 \right) \nabla^{\beta} \left[ \hat{x} , \left[ \hat{x} , \left\{ i \varepsilon \nabla , \omega_N \right\} \right] \right] \right\| _{\mathfrak{S}^{2}} &\leq C N^{1/2} \sum_{j=2}^{4 + \left| \beta \right|} \varepsilon^j \left\| W_N \right\| _{H_{3 + \left| \beta \right|}^{j}} , \end{align} $$
$$ \begin{align} \varepsilon^{\left| \beta \right|} \left\| \left( 1 + x^2 \right) \nabla^{\beta} \left[ \hat{x} , \left[ \hat{x} , \left\{ i \varepsilon \nabla , \omega_N \right\} \right] \right] i \varepsilon \nabla \right\| _{\mathfrak{S}^{2}} &\leq C N^{1/2} \sum_{j=2}^{5 + \left| \beta \right|} \varepsilon^j \left\| W_N \right\| _{H_{4 + \left| \beta \right|}^{j}} ,\end{align} $$
$$ \begin{align} \varepsilon^{\left| \beta \right|} \left\| \left( 1 + x^2 \right) \nabla^{\beta} \left[ \hat{x} , \left[ i \varepsilon \nabla , \omega_N \right] \right] \right\| _{\mathfrak{S}^{2}} &\leq C N^{1/2} \sum_{j=2}^{4 + \left| \beta \right|} \varepsilon^j \left\| W_N \right\| _{H_{2 + \left| \beta \right|}^{j}} , \end{align} $$
$$ \begin{align} \varepsilon^{\left| \beta \right|} \left\| \left( 1 + x^2 \right) \nabla^{\beta} \left[ \hat{x} , \left[ i \varepsilon \nabla , \omega_N \right] \right] i \varepsilon \nabla \right\| _{\mathfrak{S}^{2}} &\leq C N^{1/2} \sum_{j=2}^{5 + \left| \beta \right|} \varepsilon^j \left\| W_N \right\| _{H_{3 + \left| \beta \right|}^{j}}. \end{align} $$
Proof. In the following, we use
$\mathcal {W}(\omega _N)$
to denote the Wigner transform of an operator
$\omega _N$
. Due to (132) and
$$ \begin{align} \left\| \omega_{N,s} \right\| _{\mathfrak{S}^2} &= \left( \frac{2 \pi}{\varepsilon} \right)^{3/2} \left\| \mathcal{W}(\omega_{N,s}) \right\| _{L^2(\mathbb{R}^6)} \end{align} $$
the inequalities in the lemma can be obtained by estimating the
$L^2(\mathbb {R}^6)$
-norm of the respective Wigner transforms. Concerning the first inequality, we note that
$$ \begin{align} \left\| \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \textsf{D} \omega_N i \varepsilon \nabla \right\| _{\mathfrak{S}^2}&\leq \left\| \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \textsf{D} \omega_N \textsf{D} \right\| _{\mathfrak{S}^2} \end{align} $$
and
$$ \begin{align} \left\| \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \textsf{D} \omega_N \right\| _{\mathfrak{S}^2} &\leq \left\| \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \textsf{D} \omega_N \textsf{D} \right\| _{\mathfrak{S}^2} \end{align} $$
because
$\left \| i \varepsilon \nabla \textsf {D}^{-1} \right \| _{\mathfrak {S}^{\infty }} \leq 1$
and
$\left \| \textsf {D}^{-1} \right \| _{\mathfrak {S}^{\infty }} \leq 1$
. Since
$$ \begin{align} \mathcal{W} \left( \left( 1 + \hat{x}^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \widetilde{\omega}_{N,s}\right) &= \left( 1 + \frac{1}{4} \left( i \varepsilon \nabla_v + 2 x \right)^2 \right) \left( 1 + \frac{1}{4} \left( i \varepsilon \nabla_x + 2 v \right)^2 \right) \widetilde{W}_{N,s}(x,v) \nonumber \\ \mathcal{W}(\textsf{D} \widetilde{\omega}_{N,s} \textsf{D})(x,v) &= \left( 1 + \left( - i \varepsilon \nabla_x + 2 v \right)^2 \right)^{1/2} \left( 1 + \left( - i \varepsilon \nabla_x - 2 v \right)^2 \right)^{1/2} \widetilde{W}_{N,s}(x,v) , \nonumber \\ \mathcal{W} \left( \left[ \nabla, \widetilde{\omega}_{N,s} \right] \right)(x,v) &= \nabla_x \widetilde{W}_{N,s}(x,v) , \nonumber \\ \mathcal{W} \left( \left[ \hat{x} , \widetilde{\omega}_{N,s} \right] \right)(x,v) &= i \varepsilon \nabla_v \widetilde{W}_{N,s}(x,v) , \nonumber \\ \mathcal{W} \left( \widetilde{\omega}_{N,s} i \nabla \right)(x,v) &= - \frac{1}{2 \varepsilon} \left( 2 v + i \varepsilon \nabla_x \right) \widetilde{W}_{N,s}(x,v) \end{align} $$
we have
$$ \begin{align} &\mathcal{W} \left( \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right)(x,v) \nonumber \\ &\quad = \left( 1 + \frac{1}{4} \left( i \varepsilon \nabla_v + 2 x \right)^2 \right) \left( 1 + \frac{1}{4} \left( i \varepsilon \nabla_x + 2 v \right)^2 \right) \left( 1 + \left( - i \varepsilon \nabla_x + 2 v \right)^2 \right)^{1/2} \nonumber \\ &\qquad \qquad \qquad \times \left( 1 + \left( - i \varepsilon \nabla_x - 2 v \right)^2 \right)^{1/2} \widetilde{W}_{N,s}(x,v) , \end{align} $$
$$ \begin{align} &\mathcal{W} \left( \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \left[ \hat{x} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right)(x,v) \nonumber \\ &\quad = \left( 1 + \frac{1}{4} \left( i \varepsilon \nabla_v + 2 x \right)^2 \right) \left( 1 + \frac{1}{4} \left( i \varepsilon \nabla_x + 2 v \right)^2 \right) i \varepsilon \nabla_v \left( 1 + \left( - i \varepsilon \nabla_x + 2 v \right)^2 \right)^{1/2} \nonumber \\ &\qquad \qquad \qquad \times \left( 1 + \left( - i \varepsilon \nabla_x - 2 v \right)^2 \right)^{1/2} \widetilde{W}_{N,s}(x,v) , \end{align} $$
$$ \begin{align} &\mathcal{W} \left( \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \left[ \nabla , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right)(x,v) \nonumber \\ &\quad = \left( 1 + \frac{1}{4} \left( i \varepsilon \nabla_v + 2 x \right)^2 \right) \left( 1 + \frac{1}{4} \left( i \varepsilon \nabla_x + 2 v \right)^2 \right) \nabla_x \left( 1 + \left( - i \varepsilon \nabla_x + 2 v \right)^2 \right)^{1/2} \nonumber \\ &\qquad \qquad \qquad \times \left( 1 + \left( - i \varepsilon \nabla_x - 2 v \right)^2 \right)^{1/2} \widetilde{W}_{N,s}(x,v) \end{align} $$
and
$$ \begin{align} &\mathcal{W} \left( \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \left[ \hat{x} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} i \nabla \right] \right)(x,v) \nonumber \\ &\quad = - \frac{1}{2} \left( 1 + \frac{1}{4} \left( i \varepsilon \nabla_v + 2 x \right)^2 \right) \left( 1 + \frac{1}{4} \left( i \varepsilon \nabla_x + 2 v \right)^2 \right) i \nabla_v \left( 2 v + i \varepsilon \nabla_x \right) \left( 1 + \left( - i \varepsilon \nabla_x + 2 v \right)^2 \right)^{1/2} \nonumber \\ &\qquad \quad \times \left( 1 + \left( - i \varepsilon \nabla_x - 2 v \right)^2 \right)^{1/2} \widetilde{W}_{N,s}(x,v). \end{align} $$
This leads to
$$ \begin{align} \left\| \mathcal{W} \left( \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right) \right\| _{L^2(\mathbb{R}^6)} &\leq C \sum_{j=0}^6 \varepsilon^{j} \left\| \widetilde{W}_{N,s} \right\| _{H_6^j} , \end{align} $$
$$ \begin{align} \left\| \mathcal{W} \left( \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \left[ \hat{x} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right) \right\| _{L^2(\mathbb{R}^6)} &\leq C \sum_{j=1}^7 \varepsilon^{j} \left\| \widetilde{W}_{N,s} \right\| _{H_6^{j}} , \end{align} $$
$$ \begin{align} \left\| \mathcal{W} \left( \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \left[ i \nabla , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \right] \right) \right\| _{L^2(\mathbb{R}^6)} &\leq C \sum_{j=0}^6 \varepsilon^{j} \left\| \widetilde{W}_{N,s} \right\| _{H_6^{j+1}} , \end{align} $$
$$ \begin{align} \left\| \mathcal{W} \left( \left( 1 + x^2 \right) \left( 1 - \varepsilon^2 \Delta \right) \left[ \hat{x} , \textsf{D} \widetilde{\omega}_{N,s} \textsf{D} \nabla \right] \right) \right\| _{L^2(\mathbb{R}^6)} &\leq C \sum_{j=0}^7 \varepsilon^{j} \left\| \widetilde{W}_{N,s} \right\| _{H_7^{j+1}}. \end{align} $$
The remaining relations follow by similar means.
Acknowledgments
The authors would like to thank François Golse for interesting discussions about the Vlasov–Maxwell equations. N.L., moreover, would like to thank Marco Falconi for fruitful discussions about the properties of the charge distribution within project [Reference Falconi and Leopold24].
Competing interests
The authors have no competing interests to declare.
Funding statement
Support from the Swiss National Science Foundation through the NCCR SwissMAP (N.L. and C.S.) and funding from the European Union’s Horizon 2020 research and innovation programme (N.L) through the Marie Skłodowska-Curie Action EFFECT (grant agreement No. 101024712), as well as the Swiss National Science Foundation (C.S.) through the SNSF Eccellenza project PCEFP2_181153, and the Swiss State Secretariat for Research and Innovation (C.S.) through the project P.530.1016 (AEQUA) are gratefully acknowledged.
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