Proof of Theorem 1.

The proof of this theorem is very similar to the proof of the local well-posedness for quasilinear wave equations (see, for instance, Hörmander [Reference Hörmander12], Sogge [Reference Sogge28], Racke [Reference Racke27]), as well as the more modern treatment in Ifrim-Tataru [Reference Ifrim and Tataru13]. We sketch here its main steps.

(i) Energy estimates and the quasilinear energy. A key part of the argument is played by energy estimates, which we discuss here in a simpler setting, for the inhomogeneous linear problem

(3.1) $$ \begin{align} \left\{ \begin{aligned} &(\partial_t^2-\Delta_{x})U(t,x)= \mathbf{N_1}(v, \partial V) + \mathbf{N_2}(u, \partial V) + F \\ &(\partial_t^2-\Delta_{x}+1)V(t,x)=\mathbf{N_1}(v, \partial U) + \mathbf{N_2}(u, \partial U)+G,\\ \end{aligned} \right. \end{align} $$

with initial conditions

(3.2) $$ \begin{align} \left\{ \begin{aligned} & (U,V)(0,x) = (U_0(x),V_0(x))\,,\\ & (\partial_t U,\partial_t V)(0,x) = (U_1(x), V_1(x)). \end{aligned} \right. \end{align} $$

At leading order, this system agrees with the linearized equation discussed in the next section, Section 4.

Our starting point in the proof of the energy estimates is the energy functional associated to the corresponding linear equation

$$\begin{align*}E\left(U,V\right) := {\frac{1}{2}}\int_{\mathbb{R}^2} U_t ^2 + U_x^2 +V_t^2 + V_x^2 +V^2 \, dx= \int_{\mathbb{R}^2}e_0(t,x) \, dx, \end{align*}$$

where $e_0$ is the linear energy density

$$\begin{align*}e_0(t,x)={\frac{1}{2}}[U_t ^2 + U_x^2 +V_t^2 + V_x^2 +V^2 ]. \end{align*}$$

This would be the obvious candidate for the energy functional with respect to which we would like to prove the energy estimates required for the local well-posedness result. However, the right-hand side of the equations (3.1) contains second-order derivatives of $(U,V)$ , so if one tries to prove energy bounds via this functional, there would be a loss of derivatives. This is a common issue when working with quasilinear nondiagonalisable hyperbolic systems of PDEs. To avoid this loss of derivatives, we consider a quasilinear type modification of this energy, which has the form

(3.3) $$ \begin{align} E^{quasi}(U,V):= E\left(U,V\right)+ \int_{\mathbb{R}^2}B_1(v; U, V) + B_2(u; U, V)\, dx=\int_{\mathbb{R}^2} e^{quasi}(t,x) \, dx , \end{align} $$

where the quasilinear energy density is

$$\begin{align*}e^{quasi}(t,x):=e_0(t,x)+B_1(v; U, V) + B_2(u; U, V). \end{align*}$$

Here, the trilinear forms $B_1$ and $B_2$ are associated to the null forms $\mathbf N_1$ , $\mathbf N_2$ in (1.1) in a linear fashion. Precisely, corresponding to the three bilinear forms in (1.3), we have the associated corrections

(3.4) $$ \begin{align} \left\{ \begin{aligned} & B_{0i}(w; U,V):= w_t\, U_i\, \partial V, \\ & B_{ij}(w; U, V):= w_iU_j\, \partial V - w_j \,U_i\, \partial V, \\ & B_{0}(w; U, V):= - w_x\cdot U_x\, \partial V, \end{aligned} \right. \end{align} $$

for $w=u,v$ .

We will use the above energy functional in order to study the well-posedness of (3.1) in $\mathcal H^0$ . For this, we assume that $(u,v)$ are known and that we control in a pointwise fashion their associated control parameters $(A,B)$ introduced in (1.6), (1.7). Then we have

Proof. Part (i) is trivial. For clarity, we prove (ii) in the homogeneous case (i.e., when $F,G=0$ ) and leave the minor inhomogeneous adaptation to the reader. To see what is needed for the energy estimate computation, we begin by deriving the density flux relation associated to our energy density $e^{quasi}$ . We begin with the first component of $e^{quasi}$ , which is $e_0$ :

$$\begin{align*}\partial_te_0(t,x)=\sum_{j=1}^2\partial_{x_j}\left( U_tU_j +V_tV_j\right) +U_t\Box U+V_t(\Box+1)V. \end{align*}$$

The last two terms can be expanded as follows:

(3.7) $$ \begin{align} \left\{ \begin{aligned} &U_t\Box U= U_t\left( \mathbf{N_1}(v,\partial V) + \mathbf{N_2}(u, \partial V)\right) ,\\ &V_t(\Box+1)V= V_t\left( \mathbf{N_1}(v, \partial U)+\mathbf{N_2}(u, \partial U) \right). \end{aligned} \right. \end{align} $$

Next, we turn our attention to the corrections

(3.8) $$ \begin{align} \partial_t B_i(w; U,V)=B_i(w_t; U, V)+ B_i(w; U_t, V)+ B_i(w; U, V_t),\qquad w=u,v, \quad i=\overline{1,2}. \end{align} $$

Here, we will combine the first, respectively the second, terms on both RHS in the above equations with $\partial _tB_i(v;U,V)$ , respectively with $\partial _tB_i(u;U,V)$ . We obtain that

$$\begin{align*}\left\{ \begin{aligned} & U_t \mathbf{N_1}(v, \partial V)+ V_t \mathbf{N_1}(v, \partial U) + \partial_t B_1(v; U, V)=\partial_{x}C_1(\partial v, \partial U, \partial V)+ D_1(\partial^2 v, \partial U, \partial V),\\ &U_t \mathbf{N_2}(u, \partial V)+ V_t \mathbf{N_2}(u, \partial U) + \partial_t B_2(u; U, V)=\partial_{x}C_2(\partial u, \partial U, \partial V)+ D_2(\partial^2 u, \partial U, \partial V), \end{aligned} \right. \end{align*}$$

where $C_i$ and $D_i$ are trilinear forms. Their structure is unimportant here, but will be investigated later in the next section.

Summing up all these terms, we obtain the following energy flux relation for solution to (3.1):

(3.9) $$ \begin{align} \partial_t e^{quasi}(t,x)=\sum_{j=1}^2\partial_j f_j+g, \end{align} $$

where the fluxes $f_j$ have schematically the expressions

(3.10) $$ \begin{align} f_j=U_jU_t+ V_jV_t+ C_1(\partial v, \partial U, \partial V) + C_2(\partial u, \partial U, \partial V), \end{align} $$

and the source g has the form

(3.11) $$ \begin{align} g= D_1(\partial^2 v, \partial U, \partial V)+ D_2(\partial^2 u, \partial U, \partial V). \end{align} $$

To complete the proof of the energy estimates, we use the relation (3.9) to obtain

$$\begin{align*}\frac{d}{dt}\int_0^t E^{quasi}(t, U,V)=\int _0^t g\, dx, \end{align*}$$

where it remains to bound the RHS. We bound the $\partial ^2u$ and $\partial ^2 v$ factors in g in $L^{\infty }$ by $B(t)$ . The $\partial V$ and $\partial U$ factors are bounded by the energy. Then the conclusion of the Lemma follows.

(ii) Local existence. We construct a local solution to (1.1)–(1.2) using an iteration scheme. We set

$$\begin{align*}(u^{-1}, v^{-1})\equiv 0, \end{align*}$$

and define $(u^m, v^m),\ m=0,1,\dots ,$ inductively by

(3.12) $$ \begin{align} \left\{ \begin{aligned} &(\partial^2_t - \Delta_x) u^m(t,x) = \mathbf{N_1}(v^{m-1}, \partial v^m) +\mathbf{N_2}(u^{m-1}, \partial v^m)\,, \\ &(\partial^2_t - \Delta_x + 1)v^m(t,x) = \mathbf{N_1}(v^{m-1}, \partial u^m)+ \mathbf{N_2}(u^{m-1}, \partial u^m)\,, \end{aligned} \right. \ \ (t,x)\in \left[0,+\infty \right) \times \mathbb{R}^2, \end{align} $$

with

(3.13) $$ \begin{align} \left\{ \begin{aligned} & (u^m,v^m)(0,x) = (u_0(x),v_0(x))\,,\\ & (\partial_t u^m,\partial_t v^m)(0,x) = (u_1(x), v_1(x)). \end{aligned} \right. \end{align} $$

We assume the data to be in $\mathcal {S}$ so that, by the local existence theorem for linear equations, the above system admits a $\mathcal {C}^\infty $ solution for every m. We can later remove this assumption by an approximation argument.

To set the notations, we assume that at the initial time $t=0$ , we have the Lipschitz bound

(3.14) $$ \begin{align} A(0):=\sum_{\vert \alpha \vert =1}\Vert \partial^{\alpha} u(0)\Vert_{L^{\infty}} +\sum_{\vert \alpha \vert =1}\Vert \partial^{\alpha}v(0)\Vert_{L^{\infty}} \le \delta \ll 1, \end{align} $$

as well as the Sobolev bound

(3.15) $$ \begin{align} \|(u, v)[0]\|_{{\mathcal H }^n}\le M. \end{align} $$

Then we claim that there exists a time $T_0$ sufficiently small depending on $\delta $ and M such that the following bounds for the sequence $ (u^m, v^m)$ hold for all $t\in [0, T_0]$ :

(3.16) $$ \begin{align} A^{m}(t):=\sum_{\vert \alpha \vert =1}\Vert \partial^{\alpha} u^{m}(t)\Vert_{L^{\infty}} +\sum_{\vert \alpha \vert =1}\Vert \partial^{\alpha}v^{m}(t)\Vert_{L^{\infty}} \leq 2\delta, \end{align} $$

as well as the uniform energy bounds

(3.17) $$ \begin{align} \|(u^m(t), v^m(t))\|_{{\mathcal H }^n}\le 2 M \qquad \forall \ 0\le t\le T_0,\ \forall m\ge 0. \end{align} $$

When $m=0$ , the functions $(u^0, v^0)$ solve the linear system (1.4), for which the energy is conserved. Bound (3.17) is hence trivially satisfied. As for $A^0$ , we use Sobolev embeddings

$$\begin{align*}A^0( t) \leq A^0(0) + \int_{0}^t \|(\partial u^0_t,\partial v^0_t)(s)\|_{L^\infty} \, ds \leq \delta + CMT_0, \end{align*}$$

where C is some positive universal constant. Then we can choose and $T_0$ small enough (e.g., $T_0<\delta /(CM)$ ) to obtain (3.16).

Let us now suppose that our claim holds true for $m-1$ , $m\ge 1$ , and prove it for index m.

To measure $\|(u^m(t), v^m(t))\|_{{\mathcal H }^n}^2$ , we will use the energies

(3.18) $$ \begin{align} E^{quasi, n}(t, u^m, v^m)= \sum_{|\alpha| \leq n} E^{quasi}_{m-1}(t, D^\alpha_x u^m, D^\alpha_x v^m), \end{align} $$

where $E^{quasi}_{m-1}$ is obtained from $E^{quasi}$ by substituting $(u,v)$ with $(u^{m-1},v^{m-1})$ . Thanks to the smallness assumption on $A^{m-1}$ , the bound (3.5) holds for $E^{quasi}_{m-1}$ , so we will harmlessly replace $ \|(u^m, v^m)[t]\|_{{\mathcal H }^n}^2$ by $E^{quasi,n}(t, u^m, v^m)$ in (3.15) and (3.17).

We start by differentiating the system (3.12). For any $0 \leq k\le n$ , the differentiated variables $(\partial ^k_x u^m, \partial ^k_x v^m)$ solve the following system:

$$\begin{align*}\left\{ \begin{aligned} &(\partial^2_t - \Delta_x) \partial^k_x u^m(t,x) = \mathbf{N_1}(v^{m-1}, \partial \partial^k_x v^m) + \mathbf{N_2}(u^{m-1}, \partial \partial^k_x v^m) +\mathbf F_k \\ &(\partial^2_t - \Delta_x + 1) \partial^k_x v^m(t,x) = \mathbf{N_1}(v^{m-1}, \partial \partial^k_x u^m)+\mathbf{N_2}(u^{m-1}, \partial \partial^k_x u^m) +\mathbf G_k\,, \end{aligned} \right. \end{align*}$$

where $\mathbf {F}_k$ and $\mathbf {G}_k$ are given by

$$\begin{align*}\left\{ \begin{aligned} &\mathbf{F}_k:= \sum_{\substack{i+j=k\\ j<k}} \mathbf{N_1}(\partial^i_x v^{m-1}, \partial \partial^j_x v^m) + \sum_{\substack{i+j=k\\ j<k}} \mathbf{N_2}(\partial^i_x u^{m-1}, \partial \partial^j_x v^m)\\ & \mathbf{G}_k:=\sum_{\substack{i+j=k\\ j<k}} \mathbf{N_1}(\partial^i_x v^{m-1}, \partial \partial^j_x u^m)+ \sum_{\substack{i+j=k\\ j<k}} \mathbf{N_2}(\partial^i_x u^{m-1}, \partial \partial^j_x u^m). \end{aligned} \right. \end{align*}$$

We seek to apply Lemma 3.1 for this system. By Sobolev embeddings, we control

$$\begin{align*}B^{m-1}:= B(u^{m-1},v^{m-1}) \lesssim M , \end{align*}$$

and therefore, by (3.6), we have

(3.19) $$ \begin{align} \frac{d}{dt}E^{quasi}_{m-1}(\partial^k_x u^m, \partial^k_x v^m) \lesssim M \| (\partial^k_x u^m, \partial^k_x v^m)\|_{{\mathcal H }^0}^2 + \| (\partial^k_x u^m, \partial^k_x v^m)\|_{{\mathcal H }^0} \|(\mathbf F_k, \mathbf G_k)\|_{L^2}. \end{align} $$

We claim that $(\mathbf F_k, \mathbf G_k)$ can be estimated as follows:

(3.20) $$ \begin{align} \|(\mathbf F_k, \mathbf G_k)\|_{L^2} \lesssim M \|(u^m,v^m)\|_{{\mathcal H }^n}, \qquad k \leq n. \end{align} $$

Indeed, using Hölder inequality and the Gagliardo-Niremberg interpolation inequality, we see that

(3.21) $$ \begin{align} \begin{aligned} \| \mathbf{N}(\partial^i_x v^{m-1}, \partial \partial^j_x v^m)\|_{L^2} &\le \|\partial \partial^i_x v^{m-1}\|_{L^{\frac{2(k-1)}{i-1}}} \|\partial \partial^{j+1}_x v^m \|_{L^{\frac{2(k-1)}{j}}} \\ & \le \|\partial \partial^k_x v^{m-1}\|^{\alpha}_{L^2}\|\partial^2 v^{m-1}\|^{1-\alpha}_{L^\infty} \|\partial \partial^{k}_x v^{m}\|^{\beta}_{L^2}\|\partial^2 v^{m}\|^{1-\beta}_{L^\infty} \end{aligned} \end{align} $$

with $\alpha =\frac {i-1}{k-1}, \beta =\frac {j}{k-1}$ , and by Sobolev embeddings,

$$\begin{align*}\begin{aligned} & \| \mathbf{N}(\partial^i_x v^{m-1}, \partial \partial^j_x v^m)\|_{L^2}\\ & \hspace{10pt} \le \|(u^{m-1}, v^{m-1})(\tau)\|^\alpha_{{\mathcal H }^k} \|(u^{m-1}, v^{m-1})(\tau)\|^{1-\alpha}_{{\mathcal H }^3} \|(u^{m}, v^{m})(\tau)\|^\beta_{{\mathcal H }^k} \|(u^{m}, v^{m})(\tau)\|^{1-\beta}_{{\mathcal H }^3}\\ & \hspace{10pt} \le \|(u^{m-1}, v^{m-1})(\tau)\|_{{\mathcal H }^n} \|(u^{m}, v^{m})(\tau)\|_{{\mathcal H }^n}. \end{aligned} \end{align*}$$

This estimate applies for $\mathbf {N}=\mathbf {N_1}$ , and also for $\mathbf {N}=\mathbf {N_2}$ where v can be freely replaced by u. This proves (3.20). We substitute (3.20) in (3.22) and sum over $k \leq n$ . Then we obtain the energy relation

$$\begin{align*}\frac{d}{dt} E^{quasi,n}(u^m,v^m) \lesssim M \| (u^m,v^m)\|_{{\mathcal H }^n}^2 \approx M E^{quasi,n}(u^m,v^m), \end{align*}$$

and by Gronwall’s lemma,

$$\begin{align*}E^{quasi,n}(t, u^m, v^m)\le e^{CMt} E^{quasi}(0, u^m, v^m) \qquad \forall \ 0\le t\le T_0 \end{align*}$$

for some positive constant C. Then we can choose and $T_0$ small enough (e.g., $T_0<1/(2CM)$ ) to obtain (3.17).

However, the uniform bound of $(\partial u^m,\partial v^m)$ is proved as for the case $m=0$ using Sobolev embeddings,

$$\begin{align*}A^m( t) \leq A^m(0) + \int_{0}^t \|(\partial u^m_t,\partial v^m_t)(s)\|_{L^\infty} \, ds \leq \delta + 2CMT_0, \end{align*}$$

which proves that $A^m(t) \leq 2\delta $ if $T_0$ is chosen small enough.

The remaining step is to prove the convergence of the sequence of approximate solutions $(u^m, v^m)$ as $m\rightarrow \infty $ . As the problem is quasilinear, the convergence can only be shown in a weaker topology. It is enough for our goal to show that $(u^m-u^{m-1}, v^m-v^{m-1})$ is a Cauchy sequence in $C^0([0,T_0]; {\mathcal H }^0)$ . The limit $(u,v)$ will hence automatically belong to $\mathcal {H}^n$ and satisfy (1.1) together with the uniform in time bound

$$\begin{align*}\|(u,v)(t)\|_{{\mathcal H }^n}\le 2M \quad \forall 0\le t\le T_0. \end{align*}$$

From (3.12), we see that the differences $(\tilde {u}^m,\tilde {v}^m) = (u^m-u^{m-1}, v^{m}-v^{m-1})$ solve the following Cauchy problem:

$$\begin{align*}\left\{ \begin{aligned} &\Box \tilde{u}^m(t,x) = \mathbf{N_1}(v^{m-1}, \partial \tilde{v}^{m}) + \mathbf{N_2}(v^{m-1}, \partial \tilde{v}^{m}) + \mathbf{N_1}(\tilde{v}^{m-1}, \partial v^{m}) + \mathbf{N_2}(\tilde{v}^{m-1}, \partial v^{m}) \,, \\ &(\Box + 1)\tilde{v}^m(t,x) = \mathbf{N_1}(v^{m-1}, \partial \tilde{u}^{m}) + \mathbf{N_2}(v^{m-1}, \partial \tilde{u}^{m}) + \mathbf{N_1}(\tilde{u}^{m-1}, \partial v^{m}) + \mathbf{N_2}(\tilde{u}^{m-1}, \partial v^{m}) \end{aligned} \right. \end{align*}$$

with initial data $(\tilde {u}^m,\tilde {v}^m)[0] = 0$ .

We now view the last two terms in each equation as source terms and apply Lemma 3.1 to obtain

$$\begin{align*}\frac{d}{dt} E^{quasi}_{m-1} (\tilde{u}^m,\tilde{v}^m) \lesssim M (\| (\tilde{u}^m,\tilde{v}^m)\|_{{\mathcal H }^0}^2 + \| (\tilde{u}^m,\tilde{v}^m)\|_{{\mathcal H }^0} \| (\tilde{u}^{m-1},\tilde{v}^{m-1})\|_{{\mathcal H }^0}). \end{align*}$$

Then by Gronwall’s inequality, we obtain

$$ \begin{align*} &\|(\tilde{u}^m, \tilde{v}^m)(t)\|_{{\mathcal H }^0} \le CM e^{CMT_0}\int_0^t \|(\tilde{u}^{m-1}, \tilde{v}^{m-1})(\tau)\|_{{\mathcal H }^0}\, d\tau \end{align*} $$

for all $0\le t \le T_0$ . By iteration,

$$\begin{align*}\begin{aligned} \|(\tilde{u}^m , \tilde{v}^m)(t)\|_{{\mathcal H }^0} &\lesssim (CM)^m e^{mCMT_0} \int_{0\le \tau_1\le \tau_2 \le \dots\le \tau_m\le t} \|(u^0, v^0)(\tau_1)\|_{{\mathcal H }^0}\, d\tau_1 \dots d\tau_m \\ & \le \frac{(CMt)^m}{m!}e^{mCMT_0}\sup_{t\in[0,T_0]}\|(u^0, v^0)(t)\|_{{\mathcal H }^0}, \end{aligned} \end{align*}$$

which implies that the series of general term $(u^m, v^m)$ converges in $\mathcal {C}^0([0,T_0]; {\mathcal H }^0)$ and concludes the proof of the existence part $(a)$ of the theorem.

(iii) Uniqueness of solutions. This follows by the same arguments as above. We assume we have two solutions of (1.1), $(u^1,v^1)$ , $(u^2,v^2)$ , we subtract them, and we obtain a similar system as above for the difference $(\tilde {u}, \tilde {v}):=(u^1,v^1)-(u^2,v^2)$ with zero Cauchy data. Then we apply the energy estimates in Lemma 3.1 followed by Gronwall’s inequality to show $(\tilde {u},\tilde {v})=0$ . For more details, see the proof in (iv) below which yields a stronger result.

(iv) Uniform finite speed of propagation. Here, we consider two solutions of (1.1), $(u^1,v^1)$ and $(u^2,v^2)$ . We assume that their initial data coincide in a ball $B(x_0,R)$ , and show that the two solutions have to agree in the cone

$$\begin{align*}C = \{ 2t + |x-x_0| < R \}. \end{align*}$$

For the difference $(\tilde {u},\tilde {v})$ , we have the equation

$$\begin{align*}\left\{ \begin{aligned} &\Box \tilde{u}(t,x) = \mathbf{N_1}(v^{1}, \partial \tilde{v}) + \mathbf{N_2}(u^{1}, \partial \tilde{v}) + \mathbf{N_1}(\tilde{v}, \partial v^{2}) + \mathbf{N_2}(\tilde{u}, \partial v^2) \,, \\ &(\Box + 1)\tilde{v}(t,x) = \mathbf{N_1}(v^1, \partial \tilde{u}) + \mathbf{N_2}(u^1, \partial \tilde{u}) + \mathbf{N_1}(\tilde{v}, \partial u^2) + \mathbf{N_2}(\tilde{u}, \partial u^{2}). \end{aligned} \right. \end{align*}$$

We view the last two terms on the right as source terms and the rest as the equation (3.1) with $(u,v) = (u^1,v^1)$ and $(U,V)= (\tilde {u},\tilde {v})$ . Then the energy flux relation (3.9) remains valid, with the contribution of the source terms included in $D_1$ and $D_2$ in (3.11).

We integrate the energy flux relation (3.9) on the cone section $C_{[0,t_0]}=C\cap [0,t_0]$ to obtain

$$\begin{align*}\int_{C_{t_0}} e^{quasi}\, dx = \int_{C_0} e^{quasi}\, dx + \int_{C_{[0,t_0]}} g\, dx dt + F, \end{align*}$$

where the flux F is an integral over the lateral surface of the cone section which we denote by $\partial C_{[0,t]}$ ,

$$\begin{align*}F = \int_{\partial C_{[0,t_0]}} - e^{quasi} + {\frac{1}{2}} \frac{(x-x_0)_j}{|x-x_0|} f_j \, dx. \end{align*}$$

Since $A \ll 1$ , it easily follows that the contribution of the cubic terms to F is negligible and then that $F \leq 0$ . Then

$$\begin{align*}\int_{C_{t_0}} e^{quasi} \, dx \leq \int_{C_0} e^{quasi} \, dx + B \int_{ C_{[0,t_0]}} e^{quasi} \, dx dt. \end{align*}$$

At the initial time $t = 0$ , we have $(\tilde {u},\tilde {v}) = 0$ , so by Gronwall’s inequality, we obtain $e^{quasi} = 0$ inside C, which gives $(\tilde {u},\tilde {v}) = 0 $ in C.

(v) Continuation of the solution. We start by differentiating the system (3.12). For any $0 \leq k\le n$ , the differentiated variables $(\partial ^k_x u, \partial ^k_x v)$ solve the following system:

$$\begin{align*}\left\{ \begin{aligned} &(\partial^2_t - \Delta_x) \partial^k_x u(t,x) = \mathbf{N_1}(v, \partial \partial^k_x v) + \mathbf{N_2}(u, \partial \partial^k_x v) +\mathbf F_k \\ &(\partial^2_t - \Delta_x + 1) \partial^k_x v(t,x) = \mathbf{N_1}(v, \partial \partial^k_x u)+\mathbf{N_2}(u, \partial \partial^k_x u) +\mathbf G_k\,, \end{aligned} \right. \end{align*}$$

where $\mathbf {F}_k$ and $\mathbf {G}_k$ are given by

$$\begin{align*}\left\{ \begin{aligned} &\mathbf{F}_k:= \sum_{\substack{i+j=k\\ j<k}} \mathbf{N_1}(\partial^i_x v, \partial \partial^j_x v) + \sum_{\substack{i+j=k\\ j<k}} \mathbf{N_2}(\partial^i_x u, \partial \partial^j_x v)\\ & \mathbf{G}_k:=\sum_{\substack{i+j=k\\ j<k}} \mathbf{N_1}(\partial^i_x v, \partial \partial^j_x u)+ \sum_{\substack{i+j=k\\ j<k}} \mathbf{N_2}(\partial^i_x u, \partial \partial^j_x u). \end{aligned} \right. \end{align*}$$

We seek to apply Lemma 3.1 for this system. By (3.6), we have

(3.22) $$ \begin{align} \frac{d}{dt}E^{quasi}(\partial^k_x u, \partial^k_x v) \lesssim B \| (\partial^k_x u, \partial^k_x v)\|_{{\mathcal H }^0}^2 + \| (\partial^k_x u, \partial^k_x v)\|_{{\mathcal H }^0} \|(\mathbf F_k, \mathbf G_k)\|_{L^2}. \end{align} $$

It suffices to show that $(\mathbf F_k, \mathbf G_k)$ can be estimated as follows:

(3.23) $$ \begin{align} \|(\mathbf F_k, \mathbf G_k)\|_{L^2} \lesssim B \|(\partial^k u,\partial^k v)\|_{{\mathcal H }^0}, \qquad k \leq n. \end{align} $$

Indeed, using Hölder inequality and the Gagliardo-Niremberg interpolation inequality, we see that

(3.24) $$ \begin{align} \begin{aligned} \| \mathbf{N}(\partial^i_x v, \partial \partial^j_x v)\|_{L^2} &\le \|\partial \partial^i_x v\|_{L^{\frac{2(k-1)}{i-1}}} \|\partial \partial^{j+1}_x v \|_{L^{\frac{2(k-1)}{j}}} \\ & \le \|\partial \partial^k_x v\|^{\alpha}_{L^2}\|\partial^2 v\|^{1-\alpha}_{L^\infty} \|\partial \partial^{k}_x v\|^{\beta}_{L^2}\|\partial^2 v\|^{1-\beta}_{L^\infty} \end{aligned} \end{align} $$

with $\alpha =\frac {i-1}{k-1}, \beta =\frac {j}{k-1}$ . Since $\alpha +\beta =1$ ,

$$\begin{align*}\begin{aligned} & \| \mathbf{N}(\partial^i_x v, \partial \partial^j_x v)\|_{L^2} \le \| (\partial^2 u, \partial^2v )\|_{L^{\infty}} \|(\partial^ku, \partial^kv)\|_{{\mathcal H }^0}. \end{aligned} \end{align*}$$

This estimate applies for $\mathbf {N}=\mathbf {N_1}$ , and also for $\mathbf {N}=\mathbf {N_2}$ , where v can be freely replaced by u. This proves (3.23).

We substitute to obtain the energy relation

$$\begin{align*}\frac{d}{dt} E^{quasi}(\partial^k_xu,\partial^k_xv) \lesssim B \| (\partial^k u,\partial^kv)\|^2_{{\mathcal H }^0} \approx B E^{quasi}(\partial^k_x u,\partial^k_xv), \end{align*}$$

and by Gronwall’s lemma,

$$\begin{align*}E^{quasi}(t, \partial^k_xu, \partial^k_x v)\le e^{CBt} E^{quasi}(0, \partial^k_xu, \partial^k_x v) \qquad \forall \ 0\le t\le T \end{align*}$$

for some positive constant C.

Proof of Proposition 2.1.

The exterior region corresponds to $t \leq \frac {1+\vert x\vert }{4}$ . Because of the finite speed of propagation property, this region can be treated separately as long as $\partial u$ and $\partial v$ remain small. Precisely, fix a dyadic $R> 1$ and consider the solution $(u,v)$ to (1.1) in the region

$$\begin{align*}C_R^{out} = \left\{ R < |x| < 2R, \ 0 \leq t < \frac{1+|x|}{4} \right\}. \end{align*}$$

By the finite speed of propagation previously proved, the solution in this region is uniquely determined by the data in

$$\begin{align*}A_R = \{ R/2 \leq |x| \leq 4R \}. \end{align*}$$

In the region $A_R$ , our hypothesis guarantees that we have the bounds

(3.25) $$ \begin{align} \| (u,v)[0]\|_{{\mathcal H }^{2h}} \lesssim \epsilon, \qquad \| (\partial^2 u,\partial^2 v)[0]\|_{{\mathcal H }^0} \lesssim \epsilon R^{-2}. \end{align} $$

We first restrict the data to $A_R$ and then extend them to all ${\mathbb R}^2$ so that (3.25) still holds. To obtain a bound in $C_R^{out}$ , it suffices to solve the equation up to time $T_R = R/2$ . A local in time solution exists. For this solution, we make the bootstrap assumption

(3.26) $$ \begin{align} \| \partial(u,v)\|_{L^\infty} \leq \sqrt \epsilon, \qquad \| \partial^2(u,v)\|_{L^\infty} \leq R^{-1} \sqrt \epsilon \end{align} $$

in a time interval $[0,T]$ with $T \leq T_R$ . Applying the energy estimates in Theorem 1, we can propagate the energy bounds in (3.25) up to time T. Then we can use Sobolev embeddings to get pointwise bounds from the energy bounds.

For the first derivatives, this yields

$$\begin{align*}\|\partial u\|_{L^\infty} \lesssim \|\partial u\|_{L^2}^{\frac{1}{2}} \|\partial^3 u\|_{L^2}^{{\frac{1}{2}}} \lesssim \epsilon R^{-1}, \end{align*}$$

and similarly for v. This suffices in order to improve the bootstrap assumption.

For the second derivatives, this yields

$$\begin{align*}\|\partial^2 u\|_{L^\infty} \lesssim \log R \|\partial^3 u\|_{L^2} + R^{-2} \|\partial^2 u\|_{H^2} \lesssim \epsilon R^{-2}\log R, \end{align*}$$

and similarly for v. This again suffices in order to improve the bootstrap assumption.

As the bootstrap assumption can be improved for all $T < T_R$ , it follows that the solution $(u,v)$ extends to time $T_R$ and satisfies the above pointwise bounds. The proof of the proposition is concluded.

Proof. The difficulty we encounter here is that we do not have a good estimate at a fixed time for the time derivative of the energy in order to directly apply a Gronwall’s type inequality. To address this issue, the key idea is to obtain a ‘good’ energy inequality. We are led to consider the energy growth on dyadic time scales $[T, 2T]$ . Within such a dyadic time interval, it suffices to prove

(4.11) $$ \begin{align} \sup_{t\in [T, 2T]}E^{quasi} (U,V)(t) \le (1+\epsilon C)E^{quasi} (U, V)(T). \end{align} $$

As a preliminary step, we determine the growth of the $E^{quasi}(U,V)(t)$ on such a time interval:

(4.12) $$ \begin{align} E^{quasi}(U,V)(\tilde{T})-E^{quasi}(U,V)(T)=\int_T^{\tilde{T}}\frac{d}{dt}E^{quasi}(U,V)(t)\, dt \qquad \tilde{T}\in [T, 2T], \end{align} $$

where, after expanding, the RHS is a trilinear form integrated in space time, rather than at fixed time. To estimate this integral, that is, to get

(4.13) $$ \begin{align} \left| \int_T^{\tilde{T}}\frac{d}{dt}E^{quasi}(U,V)(t)\, dt\right| \lesssim \epsilon E^{quasi}(U,V) (T), \end{align} $$

we will first need to obtain the $L^2$ space-time bounds for U and V and their derivatives over various space-time regions relative to the distance to the cone.

To understand what is needed for the energy estimate computation, we begin by derivating the density flux relation associated to our energy density $e^{quasi}$ . We begin with the first component of $e^{quasi}$ , which is $e_0$ :

$$\begin{align*}\partial_te_0(t,x)=\sum_{j=1}^2\partial_{x_j}\left( U_tU_j +V_tV_j\right) +U_t\Box U+V_t(\Box+1)V. \end{align*}$$

The last two terms can be expanded as follows:

(4.14) $$ \begin{align} \left\{ \begin{aligned} &U_t\Box U= U_t\left( \mathbf{N_1}(v, \partial V)+\mathbf{N_1}(V, \partial v) + \mathbf{N_2}(u, \partial V)+\mathbf{N_2}(U, \partial v) \right) ,\\ &V_t(\Box+1)V= V_t\left( \mathbf{N_1}(v, \partial U)+\mathbf{N_1}(V, \partial u)+\mathbf{N_2}(u, \partial U)+\mathbf{N_2}(U, \partial u)\right). \end{aligned} \right. \end{align} $$

Next, we turn our attention to the quasilinear correction

(4.15) $$ \begin{align} \begin{aligned} \partial_t B_1(v; U,V)= & \ B_1(v_t; U, V)+ B_1(v; U_t, V)+ B_1(v; U, V_t) \\ \partial_t B_2(u; U,V)= & \ B_2(u_t; U, V)+ B_2(u; U_t, V)+ B_2(u; U, V_t). \end{aligned} \end{align} $$

Here, we will combine the first, respectively third, terms in both RHS in equations (4.14) with $\partial _tB_1(v;U,V)$ , respectively with $\partial _tB_2(u;U,V)$ . Schematically, we obtain that

$$\begin{align*}\left\{ \begin{aligned} &U_t \mathbf{N_1}(v, \partial V)+ V_t \mathbf{N_1}(v,\partial U) + \partial_t B_1(v; U, V)=\partial_{x}C_1(\partial v, \partial U, \partial V)+ D_1(\partial^2 v, \partial U, \partial V) \\ &U_t \mathbf{N_2}(u, \partial V)+ V_t \mathbf{N_2}(u, \partial U) + \partial_t B_2(u; U, V)=\partial_{x}C_2(\partial u, \partial U, \partial V)+ D_2(\partial^2 u, \partial U, \partial V), \end{aligned} \right. \end{align*}$$

where $C_1,C_2$ and $D_1,D_2$ are algebraic trilinear forms. Here, we need to take a closer look at the structure of $D_1$ and $D_2$ . Indeed, a simple direct computation shows that both of them have a null structure,

(4.16) $$ \begin{align} \left\{ \begin{aligned} D_1(\partial^2 v, \partial U, \partial V)= & \ \mathbf{N}(\partial v, U) \partial V + \mathbf{N}(\partial v, V) \partial U \\ D_2(\partial^2 u, \partial U, \partial V)= & \ \mathbf{N}(\partial u, U) \partial V + \mathbf{N}(\partial u, V) \partial U \end{aligned} \right. \end{align} $$

where $\mathbf {N}(\cdot , \cdot )$ denote null forms (i.e., linear combinations of the null forms in (4.2)). The relation for $D_2$ is important for us, while the one for $D_1$ is less critical because of the better $t^{-1}$ decay enjoyed by the Klein-Gordon component v.

We also note that $B_j$ and $C_j$ do not have a null structure in general, as it can be seen by examining (3.4). However, they are matched, and we will take advantage of this later on.

Summing up all these terms, we obtain the following energy flux relation for solution to inhomogeneous linearized problem:

(4.17) $$ \begin{align} \partial_te^{quasi}(t,x)=\sum_{j=1}^2\partial_j f_j+g, \end{align} $$

where the fluxes $f_j$ have the expressions

(4.18) $$ \begin{align} f_j=U_jU_t+ V_jV_t+ C_1(\partial v, \partial U, \partial V) + C_2(\partial u, \partial U, \partial V), \end{align} $$

and the source g is a trilinear form with null structure and has components as follows:

(4.19) $$ \begin{align} g= D_1(\partial^2 v, \partial U, \partial V)+ D_2(\partial^2 u, \partial U, \partial V). \end{align} $$

To complete the proof of the energy estimates, we will have to bound the source terms using the energy. The v-terms $D_1(\partial ^2 v, \partial U, \partial V)$ will be well-behaved because $\partial ^2 v$ has $t^{-1}$ decay, but the u-terms $D_2(\partial ^2 u, \partial U, \partial V)$ do not share this property. Instead, for these terms, we need a different idea which takes advantage of their null structure.

This leads us to Alinhac’s approach which establishes an improved version of the ‘standard’ energy inequality (by this we mean the inequality corresponding to the multiplier $\partial _t$ case); such an inequality yields, besides the usual fixed time energy bound, a bound of the (weighted) $L^2$ norm in both variables x and t of some good derivatives of $(U, V)$ in special regions.

The special regions mentioned above are exactly the sets $C_{TS}^\pm $ introduced earlier in (2.13), (2.14), (2.15) and (2.16), which provide a double dyadic decomposition of the space-time relative to the size of t and the size of $t-r$ which measures how far or close we are to the cone. To simplify the exposition, we will use the notation $C_{TS}$ as a shorthand for either $C^+_{TS}$ or $C^-_{TS}$ .

The good derivatives alluded to above are exactly the tangential derivatives relative to the cones $\{ t - r = const\}$ , or equivalently,Footnote ¹ relative to the hyperboloids $\{t^2- x^2 = const\}$ . These surfaces can be viewed as providing nearly equivalent foliations of the sets $C_{TS}^\pm $ .

Lemma 4.2. Assume the solutions $(u,v)$ to the main equations (1.1) or (2.17) satisfy the bounds (4.4)-(4.9) over the space-time regions $C_{T}$ . Then the solution $(U,V)$ of the linearized equation (4.1) satisfies

(4.20) $$ \begin{align} \sup_{1\le S\lesssim T}\int_{C_{TS}} \frac{1}{S} \left\{ \left( V_j +\frac{x_j}{r}V_t\right)^2 + \left( U_j +\frac{x_j}{r}U_t\right)^2 + V^2 \right\}\, dx dt \lesssim \sup_{t\in [T, 2T]}E^{quasi}(U, V)(t). \end{align} $$

Here, we only consider the $C_{TS}$ regions with $S \lesssim T$ , as the outer region $C_T^{out}$ is uninteresting from this perspective. Concerning the directional derivatives in the lemma, we note that they have a special structure:

Remark 4.3. The quantities appearing in (4.20) (i.e., $ V_j +\frac {x_j}{r}V_t$ and $ U_j +\frac {x_j}{r}U_t$ ) represent the derivatives of V, respectively U, in the tangential directions to the cones $C=\left \{t-r=const\right \}$ . We denote the them by

$$\begin{align*}\mathcal{T}_j= \partial_j +\frac{x_j}{r}\partial_t. \end{align*}$$

We further remark that we have the trivial bound

(4.21) $$ \begin{align} \sup_{1\le S \lesssim T}\int_{C_{TS}} \frac{1}{T} \left( |\nabla U|^2 + |\nabla V|^2 \right)dx dt \lesssim \sup_{t\in [T, 2T]}E^{quasi}(U, V)(t), \end{align} $$

which can be viewed as the natural complement of (4.20) for nontangential derivatives. In other words, (4.20) and (4.21) should be viewed as a pair. This last bound also shows that (4.20) becomes trivial if $S \approx T$ . Thus, in the proof, we will be concerned with the case $1 \leq S \ll T$ .

Closely related to the last comment, an important observation that applies in the $C_{TS}$ regions is that we can connect the vector fields $\mathcal {T}$ , tangent to the cones, to the corresponding vector fields Z, tangent to the hyperboloids that foliate both the interior or the exterior of the cone:

$$\begin{align*}H_\rho = \{ t^2 - x^2 = \pm \rho^2 \}. \end{align*}$$

Here, we consider a hyperboloid which intersects $C_{TS}^+$ provided that $\rho ^2 \approx TS$ . Since $S \ge 1$ , this in particular requires that

$$\begin{align*}T \lesssim \rho^2 \lesssim T^2. \end{align*}$$

In this setting, we note that vector fields Z and $\mathcal {T}$ are related in general via

(4.22) $$ \begin{align} Z = t\mathcal{T} - \frac{x}{r}(t-r)\partial_t \end{align} $$

and, in particular in the $C_{TS}$ regions, by

(4.23) $$ \begin{align} Z \approx T\mathcal{T} - S\partial_t. \end{align} $$

As the tangent planes to both the hyperboloids and cones are close to each other, via the estimate given in (4.21), we give an alternative statement of Lemma 4.2 in terms of the Z vector fields.

Lemma 4.4. Under the same assumptions as in Lemma 4.2, we have

(4.24) $$ \begin{align} \sup_{1 \leq S \lesssim T}\int_{C_{TS}} \frac{1}{S} \left\{ T^{-2} (|ZU|^2 + |ZU|^2) + V^2 \right\} + \frac{1}{T} (|\nabla U|^2 + |\nabla V|^2) \, dx dt \lesssim \sup_{t\in [T, 2T]}E^{quasi}(U, V)(t). \end{align} $$

Proof of Lemma 4.2.

We consider the following weighted version of the energy $E (U,V)$ ,

(4.25) $$ \begin{align} E_a\left(U,V\right) := \int_{\mathbb{R}^2}e^a e_0(t,x)\, dx, \end{align} $$

and similarly the weighted version of the quasilinear energy $E^{quasi}(U, V)$ ,

(4.26) $$ \begin{align} \begin{aligned} E_a^{quasi}(U,V)&:= \int_{\mathbb{R}^2}e^a e^{quasi}(t,x)\, dx. \\ \end{aligned} \end{align} $$

Here, $e^a$ is a ghost weight, which will be chosen such that a is bounded and the weight $e^a$ ultimately disappears from the inequalities. Precisely, we will choose a of the form

(4.27) $$ \begin{align} a (t,r): =-A (t-r), \end{align} $$

where A is a bounded nondecreasing function. Then the gain in the estimates will come from the contribution of $A^{\prime }(t-r)$ , which will be chosen to be positive.

We can further specialize the choice of the function $A(t-r)$ and separately adapt it to each dyadic space-time regions $C_{TS}$ for $1 \leq S \ll T$ . Precisely, we can chose it so that

(4.28) $$ \begin{align} A^{\prime}(t-r)\approx \frac{1}{S}, \qquad \mbox{ for } \vert t- r \vert \approx S, \quad \mbox{ and } A^{\prime}(t-r) = 0 \mbox{ elsewhere. } \end{align} $$

For such functions A, we need to understand the time derivative of $E_a^{quasi} (U, V)$ . Thus, using the equation (4.17), we have

$$ \begin{align*} \begin{aligned} \frac{d}{dt}E_a^{quasi}(U, V)&= \int_{\mathbb{R}^2} \frac{d}{dt} [ e^a e^{quasi} (t,x)]\, dx \\ & = \int_{\mathbb{R}^2} e^a a_t \, e^{quasi} +e^a (\partial_j f_j + g)\, dx. \\ & = \int_{\mathbb{R}^2} e^a (a_t \, e^{quasi} - a_j f_j) \, dx + \int_{\mathbb{R}^2} e^a g\, dx. \end{aligned} \end{align*} $$

The second integrand involves the function g, which is a trilinear form with a null structure. The terms in the first integrand do not separately have a null structure, so we will take a closer look at them together for our choice of a as above. Separating the quadratic and the cubic contributions, we write

$$ \begin{align*} \begin{aligned} a_t e^{quasi} - a_j f_j & = -A^{\prime}(t-r) (e^{quasi} + \frac{x_j}r f_j) \\ & = -A^{\prime}(t-r) (Q_2(\partial U,\partial V) + Q_{3,1}(\partial v,\partial U,\partial V) + Q_{3,2}(\partial u,\partial U,\partial V)) , \end{aligned} \end{align*} $$

where $Q_2$ represents the quadratic term,

$$\begin{align*}Q_2(U, V) = e_0(\partial U,\partial V) + \frac{x_j}r (U_t U_j+V_j V_t), \end{align*}$$

and $Q_{3,1},Q_{3,2}$ represent the cubic terms,

$$\begin{align*}Q_{3,i}(\partial v,\partial U,\partial V) = B_i(\partial v,\partial U,\partial V) + \frac{x}r C_i(\partial w,\partial U,\partial V). \end{align*}$$

Recombining the terms in $Q_2$ one obtains

$$\begin{align*}Q_2(U,V) = \left( V_j +\frac{x_j}{r}V_t\right)^2 + \left( U_j +\frac{x_j}{r}U_t\right)^2 + V^2 , \end{align*}$$

which is exactly as in (4.20). However, a short algebraic computation reveals the following structure for $Q_{3,i}$ :

$$\begin{align*}Q_{3,i}(\partial w,\partial U,\partial V) = D_{1,i}(\mathcal{T} w,\partial U,\partial V) + D_{2,i}(\partial w,\partial U,\partial V), \end{align*}$$

where

• • $D_{1,i}$ has no null structure but only uses a tangential derivative of w ,

• • $D_{2,j}$ has a null structure – that is, can be represented as

  $$\begin{align*}D_{2,j}(\partial w,\partial U,\partial V) =\mathbf{N}( w, U)\partial V + \mathbf{N}(w, V)\partial U. \end{align*}$$

Thus, we obtain

(4.29) $$ \begin{align} \begin{aligned} \frac{d}{dt}E_a^{quasi}(U, V) & + \int_{\mathbb{R}^2} e^a A^{\prime}(t-r) Q_2(U,V)\, dx \\ =& - \int_{\mathbb{R}^2} e^a A^{\prime}(t-r)( D_{1,1}(\mathcal{T} v,U,V) + D_{1,2}(\mathcal{T} u,U,V)) \, dx \\ & - \int_{\mathbb{R}^2} e^a A^{\prime}(t-r)( D_{2,1}(\partial v,\partial U,\partial V) + D_{2,2}(\partial u,\partial U,\partial V)) \, dx \\ & + \int_{\mathbb{R}^2} e^a ( D_1(\partial^2 v, \partial U, \partial V)+ D_2(\partial^2 u, \partial U, \partial V)) \, dx. \end{aligned} \end{align} $$

Now we integrate this relation between T and $2T$ . With our choice for A, the first integral in the left-hand side controls the expression on the left in Lemma 4.2. It remains to estimate the remaining terms on the right perturbatively.

1. The contributions of $D_{1,j}$ . Here, we use our bootstrap assumption to estimate

$$\begin{align*}| \mathcal{T} u |+|\mathcal{T} v| \lesssim \epsilon T^{-1}S^{\delta}, \end{align*}$$

which implies that

$$\begin{align*}|D_{1,1}(\mathcal{T} v,\partial U,\partial V)| +|D_{1,2}(\mathcal{T} u,\partial U,\partial V)| \lesssim \ \epsilon T^{-1}S^{\delta} |\partial U| |\partial V|. \end{align*}$$

Since $\vert A^{\prime }\vert \lesssim S^{-1}$ , this suffices in order to bound their contribution by the energy.

2. The v-terms in $D_{2,1}$ and $D_1$ . Their contribution is

(4.30) $$ \begin{align} \int_{T}^{2T} \int_{\mathbb{R}^2} e^a [A^{\prime}(t-r)D_{2,1}(\partial v,\partial U,\partial V) + D_{1}(\partial^2 v,\partial U,\partial V)]\, dxdt \end{align} $$

which are all bounded using (4.9) for the v-factors and the energy $E_a^{quasi}$ for the U and V terms by

(4.31) $$ \begin{align} \lesssim \epsilon \sup_{t\in [T, 2T]}E^{quasi}(U, V)(t). \end{align} $$

3. The u-terms in $D_{2,2}$ and $D_2$ . These have the form

(4.32) $$ \begin{align} \int_{T}^{2T}\int_{\mathbb{R}^2} e^a [A^{\prime}(t-r)D_{2,2}(\partial u,\partial U,\partial V) + D_{2}(\partial^2 u,\partial U,\partial V)] \, dx dt, \end{align} $$

which we need to process further. In the region $C^{out}_T$ , the pointwise bounds (4.7) and (4.8) give a $t^{-1}$ decay for both $\partial u$ and $\partial ^2 u$ , so this is identical to the case of the v terms above. It remains to consider the contribution over $C_T^{in}:= C_T\setminus C^{out}_T$ , where we will exploit the null structure of $D_{2,2}$ and $D_2$ ; see (4.16).

The key property is that all null forms can be expressed in the form

(4.33) $$ \begin{align} \mathbf{N}(\phi,\psi) = \partial \phi \cdot \mathcal{T} \psi + \mathcal{T}\phi \cdot \partial \psi, \end{align} $$

or equivalently,

(4.34) $$ \begin{align} \mathbf{N}(\phi,\psi) = \frac{1}{t} \left( \partial \phi \cdot Z \psi + Z \phi \cdot \partial \psi + (t-r) \partial \phi \cdot \partial \psi\right). \end{align} $$

We begin with $D_2$ , which contains terms of the form $ \mathbf N(\partial u, V) \partial U$ and $\mathbf {N}(\partial u, U)\partial V$ . We consider the first term, as the second will be similar. By (4.33), we have

(4.35) $$ \begin{align} \mathbf N(\partial u, V) = \partial^2 u \cdot \mathcal{T} V + \mathcal{T}\partial u \cdot \partial V. \end{align} $$

For the last term, we can directly use our bootstrap assumptions in (4.5) and (4.8) to obtain the pointwise bounds

$$\begin{align*}|\mathcal{T}\partial u | + \left| \frac{t-r}{t}\partial^2 u \right| \lesssim \epsilon T^{-1}S^{-\delta_1}. \end{align*}$$

Hence, the contributions of those terms are estimated as in Case 1 by (4.31).

The contribution of the first term in (4.35) to the integral in (4.32) is more delicate because now the $\mathcal {T}$ vector field applies to V. So instead, we split the integral over $C^{in}_T$ into the sum of integrals over the $C_{TS}$ space-time regions, apply the Cauchy-Schwartz inequality in space-time, and apply Hölder’s inequality in time in each such region, as well as (4.8), to bound each of these integrals by

$$\begin{align*}\begin{aligned} \left| \int_{C_{TS}} \partial U \, \partial^2 u \, \mathcal{T} V\, dx dt \right| &\lesssim \Vert \partial^2 u\Vert_{L^{\infty}_{C_{TS}}} \Vert \partial U \Vert_{L^2_{C_{TS}}} \left\| \mathcal{T} V \right\|_{L^2_{C_{TS}}} \\ &\lesssim\epsilon T^{-{\frac{1}{2}}} S^{-{\frac{1}{2}}-\delta_1} T^{{\frac{1}{2}}} \left( \sup_{t\in [T, 2T]}E^{quasi}(U, V)(t) \right)^{{\frac{1}{2}}} S^{{\frac{1}{2}}} \left\| S^{-{\frac{1}{2}}} \mathcal{T} V \right\|_{L^2_{C_{TS}}} \\ &\lesssim \epsilon S^{-\delta_1} \left( \sup_{t\in [T, 2T]}E^{quasi}(U, V)(t) \right)^{{\frac{1}{2}}} \left\| S^{-{\frac{1}{2}}} \mathcal{T} V\right\|_{L^2_{C_{TS}}}, \end{aligned} \end{align*}$$

and after a straightforward S summation over $1 \leq S \leq T$ , we get

$$\begin{align*}\begin{aligned} \left| \int_{ C_T^{in}} e^a D_{2}(\partial^2 u,\partial U,\partial V)\, dx dt \right| \lesssim & \ \epsilon \left( \sup_{t\in [T, 2T]}E^{quasi}(U, V)(t) \right)^{{\frac{1}{2}}} \sup_{1 \leq S \leq T} \left\| S^{-{\frac{1}{2}}} \mathcal{T} (U,V)\right\|_{L^2_{C_{TS}}}\\ & \ + \epsilon \sup_{t\in [T, 2T]}E^{quasi}(U, V)(t) .\\ \end{aligned} \end{align*}$$

The bound for the contribution of $D_{2,2}$ is similar, with the difference that the integrand is now localized to a fixed dyadic region $C_{TS}$ and that we are using the bootstrap assumption (4.7) for $\partial u$ instead of (4.8) for $\partial ^2 u$ . Using also (4.28), we obtain

$$\begin{align*}\begin{aligned} \left| \int_{ C_T^{in}}\!\!\! e^a A^{\prime}(t-r)D_{2,2}(\partial u,\partial U,\partial V) \, dx dt \right| \lesssim & \ \epsilon S^{-1+\delta} \left(\sup_{t\in [T, 2T]}E^{quasi}(U, V)(t) \!\right)^{{\frac{1}{2}}} \!\! \sup_{1 \leq S \leq T}\! \left\| S^{-{\frac{1}{2}}} \mathcal{T} (U,V)\right\|_{L^2_{C_{TS}}}\\ & \ + \epsilon S^{-1+\delta} \sup_{t\in [T, 2T]}E^{quasi}(U, V)(t), \!\! \end{aligned} \end{align*}$$

where the $S^{-1+\delta }$ gain insures the summation in S, but it is not otherwise needed in the sequel.

Overall, we have proved that

$$\begin{align*}(4.32) \lesssim \epsilon \sup_{t\in [T, 2T]}E^{quasi}(U, V)(t) +\epsilon \sup_{1\le S\lesssim T} \left\| S^{-{\frac{1}{2}}} \mathcal{T}(U,V)\right\|^2_{L^2_{C_{TS}}}. \end{align*}$$

Summing all up all the contributions to the integrated form of (4.29), we obtain

$$\begin{align*}\int_{C_{TS}} S^{-1}Q_2(U,V) \, dx dt \lesssim \epsilon\sup_{t\in [T, 2T]}E^{quasi}(U, V)(t) +\epsilon \sup_{1\le S\lesssim T} \left\| S^{-{\frac{1}{2}}} \mathcal{T}(U,V)\right\|^2_{L^2_{C_{TS}}}. \end{align*}$$

Finally we take the supremum over $1 \leq S \leq T$ . Then the last term on the right can be absorbed on the left, which concludes the proof of the lemma.

Now we conclude the proof of the Proposition 4.1. For this, we repeat the computation above with $a(t,r)=0$ . We integrate the relation (4.29) from T up to an arbitrary $t\in [T, 2T]$ , and estimate the RHS exactly as in the proof of the Lemma 4.4. We obtain

(4.36) $$ \begin{align} E^{quasi}_a(t) -E^{quasi}_a(T) \lesssim \epsilon \sup_{t_0\in [T, 2T]} E^{quasi}_a(t_0), \end{align} $$

and taking the supremum over $t\in [T, 2T]$ gives

$$\begin{align*}\sup_{t\in [T,2T]}E^{quasi}_a(t)\lesssim E^{quasi}_a(T), \end{align*}$$

which concludes the proof of the proposition.

Proof. With small differences, the proof mimics the proof of Proposition 4.1. We consider the domain

$$\begin{align*}D = \{ (x,t) \in C_T; \ t^2 - x^2 \leq \rho^2 \}, \end{align*}$$

which represents the portion of $C_T$ below the hyperboloid $H_\rho $ (see Figure 2, which depicts the case when the hyperboloid intersects the surface $t=2T$ , but not the $t=T$ ). Then we integrate the relation (4.17) over D, estimating the contribution of g exactly as in the proof of Proposition 4.1. The contributions on the bottom $t = T$ and the top $t = 2T$ are simply the energies. This yields the bound

(4.39) $$ \begin{align} \int_{H_\rho \cap C_T} e^{quasi}(t,x) - \frac{x_j}{t} f_j\, dx \lesssim E(U,V)(T). \end{align} $$

Here, we have used the normal vector $n = (1,-\frac {x}{t})$ on $H_{\rho }$ . It remains to verify that the integrand is positive definite and controls the integrand in the left-hand side of (4.38).

Figure 2

Region D in 1+1 space-time dimension.

The leading contribution comes from $e_0$ and the quadratic part $f_{j0}$ of the $f_j$ and gives exactly the correct expressions. It remains to perturbatively estimate the cubic contributions to the integrand in (4.39), which under the assumption (4.37), has size

$$\begin{align*}\lesssim (|\nabla v|+|\nabla u|) |\nabla U| |\nabla V| \lesssim \frac{\epsilon}{\sqrt{ST}} (|\nabla U|^2+ |\nabla V|^2) \ll \epsilon\frac{\rho^2}{T^2} (|\nabla U|^2+ |\nabla V|^2), \end{align*}$$

as needed.

Proof. The proof of the corollary is a direct consequence of Proposition 4.1 and of the variation of parameters principle (i.e., Duhamel’s principle).

Proof. The proof heavily uses the energy estimates for the linearized system (4.1) in the previous section. We will use the differentiated equations (5.1) to inductively prove bounds on the differentiated functions $(u^n,v^n)$ . For this, we will rely on the energy $E^{quasi}$ and on the bounds in Corollary 4.7.

We begin by discussing the case when $n = 0$ . The easy way to handle this case is to relate it to the linearized equation, by simply thinking at (1.1) as being written as in Corollary 4.7 where $\mathbf {F}$ and $\mathbf {G}$ are given by

$$\begin{align*}\mathbf{F}:= -\mathbf{N_1}(V,\partial v)-\mathbf{N_2}(U, \partial v) , \qquad \mathbf{G}:= -\mathbf{N_1}(V,\partial u)-\mathbf{N_2}(U, \partial u). \end{align*}$$

The non-homogeneous term $\mathbf {F}$ can be easily bound in $L^1_tL^2_{x}(C_{TS})$ using a priori estimate (4.9), while $\mathbf {G}$ is bounded in $L^1_tL^2_{x}(C_{TS})$ using the null structure highlighted in (4.33), (4.34) as in the proof of Lemma 4.4.

The case $n = 1$ is a trivial consequence as $(\nabla u, \nabla v)$ exactly solve the linearized equation. So from here on, we will assume that $n \geq 2$ .

Since we do not have a way to prove a good energy estimate working only at fixed time, we will focus on proving a good energy estimate on dyadic time scales $[T,2T]$ . Precisely, arguing by induction on n, it suffices to show that

(5.3) $$ \begin{align} \sup_{t\in [T,2T]} E^{quasi}(u^n,v^n)(t)\leq E^{quasi}(u^n,v^n)(T) + \epsilon C E^{quasi}(u^{\leq n},v^{\leq n})(T). \end{align} $$

Just as in Proposition 4.1, to prove this, we will use the stronger auxiliary norm $X^T$ in (4.40), which captures more of the structure of linearized waves. So instead of (5.3), we will prove the pair of bounds

(5.4) $$ \begin{align} \sup_{t\in [T,2T]} E^{quasi}(u^n,v^n)(t)\leq E^{quasi}(u^n,v^n)(T) + \epsilon C \|(u^{\leq n},v^{\leq n})\|_{X^{T}}^2. \end{align} $$

(5.5) $$ \begin{align} \|(u^n,v^n)\|_{X^T}^2 \lesssim E^{quasi}(u^n,v^n)(T) + \epsilon C \|(u^{\leq n},v^{\leq n})\|_{X^{T}}^2. \end{align} $$

To prove both of these bounds, we rely on the results of the Corollary 4.7, which shows that it suffices to obtain $L^1_tL^2_x$ bounds for the non-homogeneous contributions $(\mathbf {F^n}, \mathbf {G^n})$ . Precisely, we will show the bound

(5.6) $$ \begin{align} \| (\mathbf{F^n}, \mathbf{G^n})\|_{L^1 L^2_x}^2 \lesssim C \epsilon \|(u^{\leq n},v^{\leq n})\|_{X^{T}}^2 , \end{align} $$

which by Corollary 4.7 allows us to recover the estimates (5.4), (5.5).

Analogous to the discussion of the terms in (4.30), (4.32), we need to deal with two types of terms:

1. The v-terms have the form

(5.7) $$ \begin{align} \begin{aligned} \mathbf{N}_1(v^k, \partial v^{n-k})\, , \quad k=\overline{1, n-1}, \end{aligned} \end{align} $$

which we want to bound in $L^1_tL^2_x$ . Here, we do not need to use the null structure. We discuss two possible terms:

1. a) The case when $k=1$ . In this case, we use the bound (4.9) for the $\partial ^2 v$ term. We also control $\Vert \partial v^{n}\Vert _{L^2_x}$ from the $v^n$ energy. Thus, we get fixed time bound

   $$\begin{align*}\Vert \mathbf{N}_1(\partial v, \partial v^{n-1}) \Vert_{L^2_x} \lesssim \epsilon T^{-1}\Vert \nabla v^n\Vert_{L^2_x}. \end{align*}$$
   We obtain the $L^1_tL^2_x$ bound by integrating over the time interval $[T,2T]$ .

2. b) The case when $k\geq 2$ and $n-k\geq 1$ . It suffices to estimate at fixed time

   $$\begin{align*}\Vert \mathbf{N}_1(v^k, \partial v^{n-k})\Vert_{L^2_x}, \end{align*}$$
   which is done by placing $\partial v^k$ in $L^{\frac {2(n-1)}{k-1}}$ and $\partial ^2 v^{n-k}$ in $L^{\frac {2(n-1)}{n-k}}$ . If all derivatives were spatial derivatives, then we can bound these terms at fixed time using interpolation by the energy of $v^n$ and the bounds in (4.9)
   $$\begin{align*}\begin{aligned} \Vert \mathbf{N}_1(v^k, \partial v^{n-k})\Vert_{L^2_x}&\lesssim \Vert \partial v^k \Vert_{L_x^{\frac{2(n-1)}{k-1}}}\Vert \partial^2 v^{n-k}\Vert_{L_x^{\frac{2(n-1)}{n-k}}}\\ &\lesssim \Vert \partial v^n\Vert_{L^2_x} \Vert\partial^2 v\Vert_{L^{\infty}_x}\\ & \lesssim C\epsilon T^{-1}\Vert \partial v^n\Vert_{L^2_x}. \end{aligned} \end{align*}$$
   If some of these derivatives are time derivatives, then the same argument applies with the one difference that on the right we use uniform norms over the interval $[T,2T]$ . Integrating in time over the $[T,2T]$ time interval leads to the $L^1_tL^2_x$ bound.

2. The u-terms are as follows:

(5.8) $$ \begin{align} \mathbf{N}_2(u^k, \partial v^{n-k}),\quad \mathbf{N}_1(v^k, \partial u^{n-k}), \quad \mathbf{N}_2(u^k, \partial u^{n-k}) , \quad k=\overline{1, n-1}, \end{align} $$

and also need to be bounded $L^1_tL^2_x(C_{T})$ . The second term is treated as in Case 1 for $k=1$ , so we will only consider it for $k=\overline {2,n-1}$ . The analysis of the first two terms is almost identical, so we just discuss the first and third terms. In fact, we will estimate them in $L^2_tL^2_x$ and then use Hölder’s inequality:

$$\begin{align*}\Vert \mathbf{N}(u^k, \partial w^{n-k}) \Vert_{L^1_tL^2_x}\lesssim T^{{\frac{1}{2}}}\Vert \mathbf{N}(u^k, \partial w^{n-k})\Vert_{L^2_tL^2_x}, \end{align*}$$

where $ w=u,v,$ and $\mathbf {N}=\mathbf {N}_1$ or $\mathbf {N}=\mathbf {N}_2$ accordingly to (5.8). We need to estimate the nonlinearity $\mathbf {N}(u^k, \partial w^{n-k})$ in $L^2_tL^2_x$ . The difficulty here is that the second derivatives of u do not have $t^{-1}$ uniform decay; instead, they decay like $t^{-{\frac {1}{2}}}\langle t-r\rangle ^{-{\frac {1}{2}}-\delta _1}$ .

We begin by noting that in the region $C_T^{out}$ , the second derivatives of u do have $t^{-1}$ uniform decay, so again, the argument in Case 1 applies. From here on, we will consider the remaining region $C^{in}_T$ which is near the cone and corresponds to dyadic scales $ 1 \leq S \ll T$ . Here is where we make use of the null structure (4.33) as done in Lemma 4.4. We successively consider the two terms in (4.33),

(5.9) $$ \begin{align} \mathbf{N}(u^k, \partial w^{n-k}) = \partial u^k \cdot \mathcal{T} \partial w^{n-k} + \mathcal{T} u^k \cdot \partial^2 w^{n-k}. \end{align} $$

We consider the $C_{TS}$ partition of the $C_T^{in}$ and will estimate the $L^2_{t,x}(C_{ST})$ norms separately. As discussed above, we can assume that $S \ll T$ ; also, we will not distinguish between the $\pm $ (i.e., the interior vs. the exterior of the cone).

We first consider the case where $w=v$ . We estimate the first term in $C_{TS}$ using interpolation restricted to $C_{TS}$ :

$$\begin{align*}\begin{aligned} \Vert \partial u^{k}\cdot \mathcal{T} \partial v^{n-k}\Vert_{L^2}&\lesssim \Vert \partial^2 u\Vert_{L^{\infty}} ^{\frac{n-k}{n-1}} \Vert \partial u^{\leq n}\Vert_{L^2}^{\frac{k-1}{n-1}} \Vert \mathcal{T} \partial v\Vert_{L^{\infty}}^{\frac{k-1}{n-1}} \Vert \mathcal{T} v^{\leq n}\Vert_{L^2}^{\frac{n-k}{n-1}} \\ &\lesssim C \epsilon \left( T^{-{\frac{1}{2}}}S^{-{\frac{1}{2}}-\delta_1}\right)^{\frac{n-k}{n-1}} T^{-{\frac{1}{2}}\frac{k-1}{n-1}} S^{{\frac{1}{2}}\frac{n-k}{n-1}} \Vert S^{-{\frac{1}{2}}}\mathcal{T} v^{\le n}\Vert_{L^2} ^{\frac{n-k}{n-1}} \sup_{t\in [T,2T]} \Vert \nabla u^{\le n}(t)\Vert^{\frac{k-1}{n-1}}_{L^2}\\ &\lesssim C \epsilon T^{-{\frac{1}{2}}} S^{-\delta_1 \frac{n-k}{n-1}} \Vert (u^{\leq n}, v^{\le n}) \Vert_{X^T} , \end{aligned} \end{align*}$$

and similarly for the second, where we also use that $S\ll T$ in the region $C_{TS}$ :

$$\begin{align*}\begin{aligned} \Vert \mathcal{T} u^k \cdot \partial^2 v^{n-k} \Vert_{L^2}& \lesssim \Vert \mathcal{T} \partial u\Vert_{L^{\infty}}^{\frac{n-k}{n-1}} \Vert \mathcal{T} u^{\leq n}\Vert_{L^2}^{\frac{k-1}{n-1}} \Vert \partial^2 v\Vert_{L^{\infty}} ^{\frac{k-1}{n-1}} \Vert \partial ^2v^{\leq n-1}\Vert_{L^2}^{\frac{n-k}{n-1}} \\ &\lesssim C \epsilon \left(T^{-1}S^{-\delta_1}\right)^{\frac{n-k}{n-1}} S^{{\frac{1}{2}}\frac{k-1}{n-1}} T^{-\frac{k-1}{n-1} + {\frac{1}{2}}\frac{n-k}{n-1}} \Vert S^{-{\frac{1}{2}}}\mathcal{T} u^{\le n}\Vert_{L^2} ^{\frac{k-1}{n-1}} \sup_{t\in [T,2T]} \Vert \nabla v^{\le n}(t)\Vert^{\frac{n-k}{n-1}}_{L^2}\\ &\lesssim C \epsilon T^{-{\frac{1}{2}}} S^{-\delta_1 \frac{n-k}{n-1}} \Vert (u^{\leq n}, v^{\le n}) \Vert_{X^T}. \end{aligned} \end{align*}$$

Note that commutators between $\mathcal {T}$ and derivatives yield extra $T^{-1}$ factors and hence give negligible contributions. The dyadic summation over S is trivial, and hence,

$$\begin{align*}\Vert \mathbf{N}(u^k, \partial v^{n-k})\Vert_{L^2_t L^2_x}\lesssim C\epsilon T^{-{\frac{1}{2}}}\Vert (u^{\le n}, v^{\le n})\Vert_{X^T}. \end{align*}$$

A similar analysis is done on the $L^2_t L^2_x$ norm of (5.9) when $w=u$ . Using interpolation restricted to $C_{TS}$ , we find for the second term that

$$\begin{align*}\begin{aligned} &\Vert \partial^2 u^{n-k}\cdot \mathcal{T} u^k\Vert_{L^2}\lesssim \Vert \mathcal{T} u^k\Vert_{L^{\frac{2(n-1)}{k-1}}} \Vert\partial^2 u^{n-k}\Vert_{L^{\frac{2(n-1)}{n-k}}} \\&\quad \lesssim \Vert \mathcal{T} \partial u\Vert_{L^{\infty}}^{\frac{n-k}{n-1}} \Vert \mathcal{T} u^{\leq n}\Vert_{L^2}^{\frac{k-1}{n-1}} \Vert \partial^2 u\Vert_{L^{\infty}} ^{\frac{k-1}{n-1}} \Vert \partial ^2u^{\leq n-1}\Vert_{L^2}^{\frac{n-k}{n-1}} \\&\quad \lesssim C \epsilon (T^{-1}S^{-\delta_1})^{\frac{n-k}{n-1}} \left( T^{-{\frac{1}{2}}}S^{-{\frac{1}{2}}-\delta_1}\right)^{\frac{k-1}{n-1}} T^{{\frac{1}{2}}\frac{n-k}{n-1}} S^{{\frac{1}{2}}\frac{k-1}{n-1}}\Vert S^{-{\frac{1}{2}}}\mathcal{T} u^{\le n}\Vert_{L^2} ^{\frac{k-1}{n-1}} \sup_{t\in [T,2T]} \Vert \nabla u^{\le n}(t)\Vert^{\frac{n-k}{n-1}}_{L^2}\\&\quad \lesssim C \epsilon T^{-{\frac{1}{2}}} S^{-\delta_1} \Vert u^{\leq n} \Vert_{X^T}. \end{aligned} \end{align*}$$

The first estimate is treated very similarly and satisfies the same estimate.

The dyadic summation over S is once again trivial, and we find

$$\begin{align*}\Vert \mathbf{N}(u^k, \partial u^{n-k})\Vert_{L^2_t L^2_x}\lesssim C\epsilon T^{-{\frac{1}{2}}}\Vert (u^{\le n}, v^{\le n})\Vert_{X^T}. \end{align*}$$

This completes the proof of (5.6) and thus of our proposition.

Proof. The bounds (5.11) and (5.13) are direct consequences of Proposition 5.1.

The bound (5.13) is only partially implicit in the proof, as one also needs to estimate the first four terms in RHS (5.1).

Proof. Simple computations show that this is indeed the case. Iterations of the following formulas for the Klainerman vector field $\Omega _{0i}$

$$ \begin{align*} \left\{ \begin{aligned} &\Omega_{0i} Q_{12}(\phi,\psi ) = Q_{12}(\Omega_{0i}\phi , \psi) + Q_{12}(\phi, \Omega_{0i}\psi) -(-1)^i Q_{0j}(\phi,\psi), \quad &i,j\in\{1,2\}, i\ne j,\\ &\Omega_{0i} Q_{0j}(\phi,\psi) = Q_{0j}(\Omega_{0i} \phi,\psi) + Q_{0j}(\phi, \Omega_{0i}\psi)+Q_{ij}(\phi, \psi), \quad &i,j\in\{1,2\},\\ &\Omega_{0i} Q_0(\phi,\psi) = Q_0(\Omega_{0i}\phi, \psi) + Q_0(\phi, \Omega_{0i}\psi), \qquad & i=1,2, \end{aligned} \right. \end{align*} $$

as well for the $\Omega _{12}$ vector field

$$ \begin{align*} \left\{ \begin{aligned} &\Omega_{12} Q_{12}(\phi,\psi) = Q_{12}(\Omega_{12} \phi,\psi) + Q_{12}(\phi, \Omega_{12}\psi), \\ &\Omega_{12} Q_{0i}(\phi,\psi) = Q_{0i}(\Omega_{12}\phi, \psi) + Q_{0i}(\phi, \Omega_{12}\psi)+(-1)^i Q_{0j}(\phi,\psi), \quad i,j\in \{1,2\}, i\ne j,\\ &\Omega_{12} Q_0(\phi,\psi) = Q_0(\Omega_{12}\phi, \psi)+ Q_0(\phi, \Omega_{12}\psi) , \end{aligned} \right. \end{align*} $$

show that indeed the quadratic nonlinearities have a null structure. Useful in our computations are also the commutator properties of individual vector fields acting on the null structures (1.3), which we list below

(6.3) $$ \begin{align} \left[\Omega_{0i}, \partial_t \right] = -\partial_i, \quad \! \left[\Omega_{0i}, \partial_j \right] = -\delta_{ij}\partial_t, \quad \! \left[\Omega_{12}, \partial_t\right]=0, \quad \left[\Omega_{12}, \partial_1\right] = \partial_2, \quad \! \left[\Omega_{12}, \partial_2\right] = -\partial_1. \end{align} $$

Proof. As a preliminary step, we remark that our initial data energy $E^{[2h]}(u,v)(0)$ is equivalent to the square of the norm in (1.18),

(6.5) $$ \begin{align} E^{[2h]}(u,v)(0) \approx \| (u,v)[0]\|_{\mathcal H^{2h}}^2 + \| x \partial_x (u,v)[0]\|_{\mathcal H^{h}}^2 + \| x^2 \partial_x^2 (u,v)[0]\|_{\mathcal H^0}^2. \end{align} $$

This is a straightforward elliptic computation which is left for the reader. We will take advantage of this observation in order to simplify the analysis in the exterior region $C^{out}$ . Indeed, in this region, we can directly apply the result of Proposition 2.2 to obtain the desired bounds on the solution, and we can dispense with the vector field analysis. Precisely, we conclude that we have the exterior fixed time uniform estimate

(6.6) $$ \begin{align} E^{[2h]}_{ext}(u,v)(t) \lesssim E^{[2h]}(u,v)(0), \end{align} $$

where

$$\begin{align*}E^{[2h]}_{ext}(u,v)(t) = \| (u,v)[t]\|_{\mathcal H^{2h}(C^{out})}^2 + \| x \partial_x (u,v)[t]\|_{\mathcal H^{h}(C^{out})}^2 + \| x^2 \partial_x^2 (u,v)[t]\|_{\mathcal H^0(C^{out})}^2. \end{align*}$$

This bound is stronger than the needed one in (6.4) in the exterior region in two ways: (i) it does not have the $t^{\tilde {C}\epsilon }$ loss, and (ii) it applies to all vector fields of size $|x|$ rather than only the Z vector fields.

After these preliminaries, we turn our attention to the evolution of the full vector field energies of $(u,v)$ . We begin with several reductions which follow the pattern of previous sections. First we recall that our problem is quasilinear and the energy that can be propagated for the derivatives of $(u,v)$ is the quasilinear energy. So denoting

(6.7) $$ \begin{align} E^{[2h]}_{quasi}(u,v):=\sum_{|\gamma| \leq 2h} E^{[2h]}_{quasi} (t; {\mathcal Z}^\gamma u, {\mathcal Z}^\gamma v), \end{align} $$

we will replace the bound (6.4) with the equivalent bound

(6.8) $$ \begin{align} E^{[2h]}_{quasi} (v, u)(t)\lesssim t^{\tilde{C}\epsilon}E^{[2h]}_{quasi}(v,u)(0), \quad t\in [0,T^{\prime}]. \end{align} $$

As before, this reduces to a bound on a dyadic time interval

(6.9) $$ \begin{align} E^{[2h]}_{quasi}(u,v)(t)\leq (1+ \tilde C\epsilon) E^{[2h]}_{quasi}(u,v)(T), \quad t\in [T,2T]\subset [0, T^{\prime}]. \end{align} $$

Applying Corollary 4.7, bound (6.9) in turn would follow from a bound for the source terms in the equation (6.1) in the time interval $[T,2T]$ , which we separate into an interior and an exterior part:

(6.10) $$ \begin{align} \| (\mathbf{F}^\gamma,\mathbf{G}^\gamma) \|_{L^1_t L^2_x(C_T^{int})} \lesssim C \epsilon \| {\mathcal Z}^{\leq 2h} (u ,v) \|_{X^T},\qquad |\gamma| \leq 2h, \end{align} $$

respectively

(6.11) $$ \begin{align} \| (\mathbf{F}^\gamma,\mathbf{G}^\gamma) \|_{L^1_t L^2_x(C_T^{out})} \lesssim C \epsilon \| {\mathcal Z}^{\leq 2h} (u ,v)[0] \|_{{\mathcal H }^0}, \qquad |\gamma| \leq 2h. \end{align} $$

This separation is convenient here because in the exterior region, we have access to the stronger bounds in (6.6) which simplifies matters somewhat. A similar separation could have been implemented in the previous two sections, but there it would have made less of a difference.

Since $(u,v)$ play symmetric roles in this analysis, we will use the notations w and ${\omega }$ for either u or v. Then we need to estimate

$$\begin{align*}\| \mathbf{N}(w^{\gamma_1}, \partial {\omega}^{\gamma_2})\|_{L^1 _tL^2_x} \lesssim C \epsilon \| {\mathcal Z}^{\leq 2h} (u,v) \|_{X^T}, \end{align*}$$

where $\gamma _1$ and $\gamma _2$ are restricted to the range

$$\begin{align*}|\gamma_1|+|\gamma_2| \leq 2h, \qquad |\gamma_1|, |\gamma_2| < 2h. \end{align*}$$

Setting $\gamma _i = (\alpha _i,\beta _i)$ , we distinguish three cases:

1. I) $ \beta _1 = \beta _2 = 0$ .

2. II) $|\beta _1| = 0$ , $|\beta _2| = 1$ or viceversa.

3. III) $|\beta _1| = |\beta _2| = 1$ .

The case I) was already considered in Section 5. For the case II), we enumerate the possibilities:

$$\begin{align*}\mathbf{N}( \partial^{n_1} w, \partial^{n_2} Z {\omega}) \qquad 1\leq n_1+ n_2 \leq h+1, \ n_2 \leq h. \end{align*}$$

The case $n_1 = 0$ may occur here only if we started with exactly $Z^2$ and one Z was commuted. In this case, we get the terms

$$\begin{align*}\mathbf{N}( w, \partial Z {\omega}), \end{align*}$$

which have not been covered yet. We will still discard this case because it is similar but simpler than case (6.14) below. The case $n_1 = 1$ is also identical to the estimates in Section 5, using directly the bootstrap assumptions for the second-order derivatives for u and v, so we can also discard it. Thus, we are left with

(6.12) $$ \begin{align} \mathbf{N}(\partial^{n_1} w, \partial^{n_2} Z{\omega}) \qquad 2 \leq n_1 \leq h, \ 1\leq n_1+n_2 \leq h+1, \end{align} $$

(6.13) $$ \begin{align} \mathbf{N}(\partial^{h+1} w, Z{\omega}). \end{align} $$

Finally, in case III), the only terms to consider are

(6.14) $$ \begin{align} \mathbf{N}(Zw,\partial Z{\omega}). \end{align} $$

Proof. Using hyperbolic polar coordinates adapted to $C_{TS}$ (see the next section), this lemma reduces to variants of the classical Gagliardo-Nirenberg inequality in a unit sized domain. The details of the proof are included in the appendix.