Local existence of Sobolev and classical solutions of a periodic symmetric hyperbolic system, uniform on bounded H^s data

(Uniformly admissible projected data.) If the stated data class is empty, choose any positive $T$ with $[-T,T]$ inside $I$ and any $C>0$; all quantified conclusions are vacuous. Otherwise $K$ is nonempty. Choose $\tau >0$ with $[-\tau ,\tau ]\subseteq I$ and $\delta >0$ with the closed $3\delta$-neighborhood of $K$ contained in $\mathrm{int}{K}^{\mathrm{\prime }}$. Let $Q$ be the closed $2\delta$-neighborhood of $K$, a compact subset of $\mathrm{int}{K}^{\mathrm{\prime }}$. The tail and embedding estimates give $\mathrm{\parallel }{P}_N{u}_0-u_0{\mathrm{\parallel }}_{C^0}\le C_{\mathrm{e}\mathrm{m}\mathrm{b}}{C}_s(1+N^2{)}^{-1\mathrm{/}2}R$. Choose $N_0$ so the right side is less than $\delta$ for every $N\ge N_0$. Thus $P_N{u}_0$ takes values in the closed $\delta$-neighborhood of $K$, hence in $Q\subseteq O$, and $\mathrm{\parallel }{P}_N{u}_0{\mathrm{\parallel }}_{H^s}\le R$. Only this cofinal family of admissible projections is used.

(Strict bootstrap constants.) On $[-\tau ,\tau ]\times {\mathbb{T}}^3\times Q$ choose positive energy-equivalence constants $a,b$ before choosing the norm radius. Choose $R_1>R$ with $a{R}_1^2>2(1+b{R}^2)$. Obtain $C_1,C_2$ for value set $Q$ and norm ball of radius $R_1$. Choose $0<T<\tau$ with $(1+b{R}^2)e^{C_{1}T}<a{R}_1^2$ and $T{C}_{\mathrm{e}\mathrm{m}\mathrm{b}}{C}_2(1+R_1)<\delta \mathrm{/}2$. Every constant is fixed before choosing $T$, and the energy inequality already has a strict margin at $T=0$.

(Uniform existence by first exit.) For $N\ge N_0$ solve the admissible ODE from $P_N{u}_0$. On the connected component containing $0$ of times with values in $\mathrm{int}Q$ and $\mathrm{\parallel }{u}_N{\mathrm{\parallel }}_{H^s}<R_1$, the absolute energy inequality gives $1+e_N(t)\le (1+e_N(0))e^{C_1\mathrm{\mid }t\mathrm{\mid }}\le (1+b{R}^2)e^{C_{1}T}<a{R}_1^2$. The derivative and embedding bounds give $\mathrm{\parallel }{u}_N(t)-P_N{u}_0{\mathrm{\parallel }}_{C^0}<\delta \mathrm{/}2$. Step 1.1 therefore places all values in the closed $3\delta \mathrm{/}2$-neighborhood of $K$, a compact subset of $\mathrm{int}Q$. These strict margins exclude a first exit before $T$ in either direction. For each fixed $N$ the norm and value bounds give a compact coordinate trajectory inside the ODE domain; since $[-T,T]\subseteq I$, continuation excludes a maximal endpoint before $T$. Thus every $N\ge N_0$ exists on $(-T,T)$ with values in $Q\subseteq K^{\mathrm{\prime }}$, norm at most $R_1$ and time derivative norm at most $C_2(1+R_1)$.

(Extraction and passage to the limit.) Apply compactness to $N=N_0,N_0+1,\dots$; its proof uses only boundedness and equicontinuity, so this relabeling gives indices $N_j\to \mathrm{\infty }$ and locally uniform $H^{s-1}$ convergence to $u$. The limit lemma gives the asserted classical Sobolev solution with datum $u_0$, values in $Q\subseteq K^{\mathrm{\prime }}$, and bounds $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}{H}^s}\le C_s{R}_1$ and $\mathrm{\parallel }{\mathrm{\partial }}_{t}u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}{H}^{s-1}}\le C_{\mathrm{l}\mathrm{i}\mathrm{m}}{C}_2(1+R_1)$, where $C_s,C_{\mathrm{l}\mathrm{i}\mathrm{m}}$ cover its Fourier norm-equivalence factors. Choose $C$ to dominate these two constants divided by $1+R$. This proves the statement.