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Optimal transport tasks naturally arise in gas networks, which include a variety of constraints such as physical plausibility of the transport and the avoidance of extreme pressure fluctuations. To define feasible optimal transport plans, we utilize a 
$p$-Wasserstein metric and similar dynamic formulations minimizing the kinetic energy necessary for moving gas through the network, which we combine with suitable versions of Kirchhoff’s law as the coupling condition at nodes. In contrast to existing literature, we especially focus on the non-standard case 
$p \neq 2$ to derive an overdamped isothermal model for gases through 
$p$-Wasserstein gradient flows in order to uncover and analyze underlying dynamics. We introduce different options for modelling the gas network as an oriented graph including the possibility to store gas at interior vertices and to put in or take out gas at boundary vertices.
In this paper, we establish suitable characterisations for a pair of functions 
$(W(x),H(x))$ on a bounded, connected domain 
$\Omega \subset \mathbb{R}^n$ in order to have the following Hardy inequality:
\begin{equation*}\int_{\Omega} W(x) |\nabla u|_A^2 dx \geq \int_{\Omega} |\nabla d|^2_AH(x)|u|^2 dx, \,\,\, u \in C^{1}_0(\Omega),\end{equation*}
where d(x) is a suitable quasi-norm (gauge), 
$|\xi|^2_A = \langle A(x)\xi, \xi \rangle$ for 
$\xi \in \mathbb{R}^n$ and A(x) is an n × n symmetric, uniformly positive definite matrix defined on a bounded domain 
$\Omega \subset \mathbb{R}^n$. We also give its Lp analogue. As a consequence, we present examples for a standard Laplacian on 
$\mathbb{R}^n$, Baouendi–Grushin operator, and sub-Laplacians on the Heisenberg group, the Engel group and the Cartan group. Those kind of characterisations for a pair of functions 
$(W(x),H(x))$ are obtained also for the Rellich inequality. These results answer the open problems of Ghoussoub-Moradifam [16].
We solve a problem posed by Calabi more than 60 years ago, known as the Saint-Venant compatibility problem: Given a compact Riemannian manifold, generally with boundary, find a compatibility operator for Lie derivatives of the metric tensor. This problem is related to other compatibility problems in mathematical physics, and to their inherent gauge freedom. To this end, we develop a framework generalizing the theory of elliptic complexes for sequences of linear differential operators 
$(A_{\bullet })$ between sections of vector bundles. We call such a sequence an elliptic pre-complex if the operators satisfy overdetermined ellipticity conditions and the order of 
$A_{k+1}A_k$ does not exceed the order of 
$A_k$. We show that every elliptic pre-complex 
$(A_{\bullet })$ can be ‘corrected’ into a complex 
$({\mathcal {A}}_{\bullet })$ of pseudodifferential operators, where 
${\mathcal {A}}_k - A_k$ is a zero-order correction within this class. The induced complex 
$({\mathcal {A}}_{\bullet })$ yields Hodge-like decompositions, which in turn lead to explicit integrability conditions for overdetermined boundary-value problems, with uniqueness and gauge freedom clauses. We apply the theory on elliptic pre-complexes of exterior covariant derivatives of vector-valued forms and double forms satisfying generalized algebraic Bianchi identities, thus resolving a set of compatibility and gauge problems, among which one is the Saint-Venant problem.
We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that includes strong existence, path-by-path uniqueness, existence of a solution flow of diffeomorphisms, Malliavin differentiability and 
$\rho $-irregularity. As a consequence, we can also treat McKean-Vlasov, transport and continuity equations.
Let 
$\Omega \subset \mathbb{R}^d$ with 
$d\geq 2$ be a bounded domain of class 
${\mathcal C}^{1,\beta }$ for some 
$\beta \in (0,1)$. For 
$p\in (1, \infty )$ and 
$s\in (0,1)$, let 
$\Lambda ^s_{p}(\Omega )$ be the first eigenvalue of the mixed local–nonlocal operator 
$-\Delta _p+(-\Delta _p)^s$ in Ω with the homogeneous nonlocal Dirichlet boundary condition. We establish a strict Faber–Krahn-type inequality for 
$\Lambda _{p}^s(\cdot )$ under polarization. As an application of this strict inequality, we obtain the strict monotonicity of 
$\Lambda _{p}^s(\cdot )$ over the annular domains and characterize the rigidity property of the balls in the classical Faber–Krahn inequality for 
$-\Delta _p+(-\Delta _p)^s$.
Let X be a complex Banach space and B be a closed linear operator with domain 
$\mathcal{D}(B) \subset X,\,\, a,b,c,d\in\mathbb{R},$ and 
$0 \lt \beta \lt \alpha.$ We prove that the problem
\begin{equation*}u(t) -(aB+bI)(g_{\alpha-\beta}\ast u)(t) -(cB+dI)(g_{\alpha}\ast u)(t) = h(t), \quad t\geq 0,\end{equation*}
where 
$g_{\alpha}(t)=t^{\alpha-1}/\Gamma(\alpha)$ and 
$h:\mathbb{R}_+\to X$ is given, has a unique solution for any initial condition on 
$\mathcal{D}(B)\times X$ as long as the operator B generates an ad-hoc Laplace transformable and strongly continuous solution family 
$\{R_{\alpha,\beta}(t)\}_{t\geq 0} \subset \mathcal{L}(X).$ It is shown that such a solution family exists whenever the pair 
$(\alpha,\beta)$ belongs to a subset of the set 
$(1,2]\times(0,1]$ and B is the generator of a cosine family or a C0-semigroup in 
$X.$ In any case, it also depends on certain compatibility conditions on the real parameters 
$a,b,c,d$ that must be satisfied.
We consider the pseudorelativistic Hartree equation 
$$ \begin{align*} i\partial_t\psi=(\sqrt{-c^2\Delta +m^2c^4}-mc^2)\psi-(|x|^{-1}*|\psi|^2)\psi\quad \text{with } (t,x)\in\mathbb{R}\times\mathbb{R}^3, \end{align*} $$
which describes the dynamics of pseudorelativistic boson stars in the mean-field limit. We study the travelling waves of the form 
$\psi (t,x)=e^{it\mu }\varphi _{c}(x-vt)$, where 
$v\in \mathbb {R}^3$ denotes the travelling velocity. We prove that 
$\varphi _{c}$ converges strongly to the minimiser 
$\varphi _{\infty }$ of the limit energy 
$E_{\infty }(N)$ in 
$H^1(\mathbb {R}^3)$ as the light speed 
$c\to \infty $, where 
$E_{\infty }(N)$ is the corresponding energy for the limit equation 
$$ \begin{align*} -\frac{1}{2m}\Delta\varphi_{\infty}+i(v\cdot\nabla)\varphi_{\infty}-({|x|^{-1}}*|\varphi_{\infty}|^2)\varphi_{\infty}=-\lambda\varphi_{\infty}. \end{align*} $$
Since the operator 
$-\Delta $ is the classical kinetic operator, we call this the nonrelativistic limit. We prove the existence of the minimiser for the limit energy 
$E_{\infty }(N)$ by using concentration-compactness arguments.
Recently, we analysed spontaneous symmetry breaking (SSB) of solitons in linearly coupled dual-core waveguides with fractional diffraction and cubic nonlinearity. In a practical context, the system can serve as a model for optical waveguides with the fractional diffraction or Bose–Einstein condensate of particles with Lévy index 
$\alpha <2$. In an earlier study, the SSB in the fractional coupler was identified as the bifurcation of subcritical type, becoming extremely subcritical in the limit of 
$\alpha \rightarrow 1$. There, the moving solitons and collisions between them at low speeds were also explored. In the present paper, we present new numerical results for fast solitons demonstrating restoration of symmetry in post-collision dynamics.
When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how the perturbation of MCMC affects the convergence speed and approximation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has fast convergence speed and high approximation accuracy as well. Our convergence analysis is conducted under either the Wasserstein metric or the 
$\chi^2$ metric, both are widely used in the literature. The results can be extended to obtain non-asymptotic error bounds for MCMC estimators. We demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis, Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally, we present some simple numerical examples to verify our theoretical claims.
We study the existence and strong instability of standing wave solutions for the fractional nonlinear Schrödinger equation
\begin{equation*}\begin{cases}\mathrm{i} \psi_{t}=\left(-\Delta\right)^s \psi-\left(|\psi|^{p-2}\psi+\mu|\psi|^{q-2}\psi\right), & \quad(t,x)\in(0,\infty)\times\mathbb{R}^N,\\\psi(0,x)=\psi_0(x),&\quad x\in\mathbb{R}^N,\end{cases}\end{equation*}
where 
$N\geq2$, 
$0 \lt s \lt 1$, 
$2 \lt q \lt p \lt 2_s^*=2N/(N-2s)$, and 
$\mu\in\mathbb{R}$. The primary challenge lies in the inhomogeneity of the nonlinearity.We deal with the following three cases: (i) for 
$2 \lt q \lt p \lt 2+4s/N$ and µ < 0, there exists a threshold mass a0 for the existence of the least energy normalized solution; (ii) for 
$2+4s/N \lt q \lt p \lt 2_s^*$ and µ > 0, we reveal the existence of the ground state solution, explore the strong instability of standing waves, and provide a blow-up criterion; (iii) for 
$2 \lt q\leq2+4s/N \lt p \lt 2_s^*$ and µ < 0, the strong instability of standing wave solutions is demonstrated. These findings are illuminated through variational characterizations, the profile decomposition, and the virial estimate.
In the present paper, we consider the following fractional double phase problem with nonlocal reaction:
\begin{align*}\qquad \quad \left\{\begin{array}{lll}(- \Delta)_p^s u + (- \Delta)_q ^s u + V(\varepsilon x) (|u|^{p - 2} u + |u|^{q - 2} u) = \left(\frac{1}{|x|^\mu} * F(u)\right) f(u) & \ \text{in}\ \mathbb{R}^N, \\u \in W^{s,p}(\mathbb{R}^N) \cap W^{s,q}(\mathbb{R}^N) ,\ u \gt 0 & \ \text{in}\ \mathbb{R}^N,\end{array}\right\}\end{align*}
where ɛ is apositive parameter, 
$0 \lt s \lt 1$, 
$2 \leqslant p \lt q \lt \min\{2p, N / s\}$, 
$0 \lt \mu \lt sp$, 
$(- \Delta)_t^s$ 
$(t \in \left\{p,q\right\})$ is the fractional t-Laplace operator, the reaction term 
$f : \mathbb{R} \mapsto \mathbb{R}$ is continuous, and the potential 
$V \in C (\mathbb{R}^N , \mathbb{R})$ satisfying a local condition. Using a variational approach and topological tools (the non-standard C1-Nehari manifold analysis and the abstract category theory), multiplicity of positive solutions and concentration properties for the above problem are established. Our results extend and complement some previous contributions related to double phase variational integrals.
We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,where 
\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}
$\phi *u$ is a spatial convolution with the top hat kernel, 
$\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$. After observing that the problem is globally well-posed, we demonstrate that positive, spatially periodic solutions bifurcate from the spatially uniform steady state solution 
$u=1$ as the diffusivity, 
$D$, decreases through 
$\Delta _1 \approx 0.00297$ (the exact value is determined in Section 3). We explicitly construct these spatially periodic solutions as uniformly valid asymptotic approximations for 
$D \ll 1$, over one wavelength, via the method of matched asymptotic expansions. These consist, at leading order, of regularly spaced, compactly supported regions with width of 
$O(1)$ where 
$u=O(1)$, separated by regions where 
$u$ is exponentially small at leading order as 
$D \to 0^+$. From numerical solutions, we find that for 
$D \geq \Delta _1$, permanent form travelling waves, with minimum wavespeed, 
$2 \sqrt{D}$, are generated, whilst for 
$0 \lt D \lt \Delta _1$, the wavefronts generated separate the regions where 
$u=0$ from a region where a steady periodic solution is created via a distinct periodic shedding mechanism acting immediately to the rear of the advancing front, with this mechanism becoming more pronounced with decreasing 
$D$. The structure of these transitional travelling wave forms is examined in some detail.
