Periodic first-order symmetric hyperbolic systems: coefficients, admissible values, data classes and solutions

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix an integer $m\ge 1$, an open set $O\subseteq {\mathbb{R}}^m$ with the subspace topology and a nondegenerate open interval $I\subseteq \mathbb{R}$ containing $0$. All functions on ${\mathbb{T}}_{2\pi }^3$ are those of The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s, and smoothness on $I\times {\mathbb{T}}_{2\pi }^3\times O$ means smoothness of the representative in the sense of clause (e) of that item.

(a) Coefficients. A periodic symmetric hyperbolic coefficient family on $O$ consists of smooth maps $A^{\mu }:I\times {\mathbb{T}}_{2\pi }^3\times O\to {\mathrm{Sym}}_m(\mathbb{R})(\mu =0,1,2,3),B:I\times {\mathbb{T}}_{2\pi }^3\times O\to {\mathbb{R}}^m,$ such that $A^0(t,x,u)$ is positive definite for every $(t,x,u)\in I\times {\mathbb{T}}_{2\pi }^3\times O$; here ${\mathrm{Sym}}_m(\mathbb{R})$ is the real vector space of symmetric $m\times m$ matrices, identified with symmetric endomorphisms of ${\mathbb{R}}^m$ through the standard coordinate inner product, and positive definiteness means $z^T{A}^0(t,x,u)z>0$ for every nonzero $z\in {\mathbb{R}}^m$ ($C^k$ maps and multi-index derivative notation in Euclidean space).

(b) The system. The associated symmetric hyperbolic system is $A^0(t,x,u){\mathrm{\partial }}_{t}u+{\sum }_{i=1}^3{A}^i(t,x,u){\mathrm{\partial }}_{i}u+B(t,x,u)=0.(\star )$

(c) Values and norms. For each integer $s\ge 0$, an $H^s$ class $u\in H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$ takes values in a set $E\subseteq {\mathbb{R}}^m$ when it has a continuous representative whose values lie in $E$. Such a representative is unique, since continuous functions equal almost everywhere on the torus agree everywhere. A map $u:J\to H^s$ with $J\subseteq I$ takes values in $E$ when every class $u(t)$ does. For compact $K\subseteq O$ and $R\ge 0$ put ${\mathcal{S}}_s(K,R):=\{u_0\in H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m):u_0\text{ takes values in }K,\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^s}\le R\}\mathrm{.}$ When an item fixes $s$, it may abbreviate this class by $\mathcal{S}(K,R)$.

(d) Classical solutions. Let $s\ge 2$ be an integer and let $J\subseteq I$ be an interval. A classical solution of $(\star )$ on $J\times {\mathbb{T}}_{2\pi }^3$ is a $C^1$ map $u:J\times {\mathbb{T}}_{2\pi }^3\to {\mathbb{R}}^m$ with $u(t,x)\in O$ for all $(t,x)$ and $(\star )$ satisfied at every point, such that $u\in L^{\mathrm{\infty }}(J;H^s)$ and ${\mathrm{\partial }}_{t}u\in L^{\mathrm{\infty }}(J;H^{s-1})$.

(e) Sobolev solutions. Let $s\ge 5$ be an integer and let $J\subseteq I$ be an open interval. A map $u:J\to H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$ is a Sobolev solution of $(\star )$ on $J$ when

1. $u\in L^{\mathrm{\infty }}(J;H^s)\cap C^0(J;H^{s-1})$ and the distributional time derivative ${\mathrm{\partial }}_{t}u$ lies in $L^{\mathrm{\infty }}(J;H^{s-1})$;
2. $u$ takes values in $O$ using the continuous representatives from (c);
3. for almost every $t\in J$ the identity $(\star )$ holds in $L^2({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$, with coefficients evaluated using the continuous representative of $u(t)$. This condition requires each displayed product to define an $L^2$ class; it does not assert a global coefficient-composition bound depending only on $\mathrm{\parallel }u(t){\mathrm{\parallel }}_{H^s}$.

(f) Initial data. For an integer $s\ge 5$, initial data for $(\star )$ at time $0$ are a class $u_0\in H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$ whose values lie in a compact subset of $O$; a solution has initial value $u_0$ when $u(t)\to u_0$ in $H^{s-1}$ as $t\to 0$ within $J$.

The definition introduces the system and its solution classes only; existence, uniqueness, continuous dependence and smooth persistence are asserted and proved in Local existence of Sobolev and classical solutions of a periodic symmetric hyperbolic system, uniform on bounded H^s data, Uniqueness of classical solutions and continuous dependence in H^{s-1} for periodic symmetric hyperbolic systems and Smooth data give smooth solutions of a periodic symmetric hyperbolic system on the existence interval.