Exact periodic positive symmetric-hyperbolic supplier

Apply [F3] to the state class and coefficient family in [F1]–[F2]. Its hypotheses match term by term: the unknown has $m$ real components on the specified $2\pi$-periodic torus, the coefficients are smooth on the same open state domain, $A^{\mu }$ are symmetric, $A^0$ is positive definite, the data range in compact $K⋐K^{\mathrm{\prime }}$, and the integer regularity is $s\ge 5$. It gives the common interval, value range, solution regularity and bounds.

(Every solution in the stated Sobolev class is classical.) Let $v$ be any Sobolev solution on $(-T,T)$ with values in $K^{\mathrm{\prime }}$. Since $s\ge 5$, the embedding [F6] with $q=s-1$ and $m=1$ applies, so $v$ and its first spatial derivatives have representatives continuous jointly in $(t,x)$; continuity in $t$ follows from $v\in C^0{H}^{s-1}$. On each compact time subinterval, the matrix $A^0(t,x,v(t,x))$ is uniformly invertible: it is positive definite and its arguments range over a compact subset of $I\times {\mathbb{T}}_{2\pi }^3\times K^{\mathrm{\prime }}$. Define the continuous function $P_v:=-(A^0(t,x,v){)}^{-1}({\sum }_i{A}^i(t,x,v){\mathrm{\partial }}_{i}v+B(t,x,v))$. The distributional equation and the a.e.-in-time $L^2$ equation in the Sobolev-solution definition imply ${\mathrm{\partial }}_{t}v=P_v$ as a spacetime distribution. For every smooth spatial test function $\phi$, the continuous scalar function $t\mapsto \int v(t,x)\phi (x)d\mu (x)$ therefore has distributional derivative $\int P_v(t,x)\phi (x)d\mu (x)$; the ordinary fundamental theorem of calculus gives $v(t)-v(t_0)={\int }_{t_0}^t{P}_v(r)dr$ in $L^2$ for every $t,t_0$. Both sides have continuous spatial representatives, so this equality holds pointwise. Thus $v$ is $C^1$ in spacetime and solves $(\star )$ pointwise. Its original $L^{\mathrm{\infty }}{H}^s$ and $L^{\mathrm{\infty }}{H}^{s-1}$ time-derivative bounds make it a classical solution in the exact sense of Periodic first-order symmetric hyperbolic systems: coefficients, admissible values, data classes and solutions.

Let $u,v$ be two Sobolev solutions in the class specified by the Statement with the same initial state. Their essential $H^s$ bounds extend to every time: for any $t$, choose times outside the exceptional null set converging to $t$; continuity in $H^{s-1}$ makes every finite Fourier partial sum of $N_s(u(t){)}^2$ the limit of the corresponding sums at those times, so it is bounded by the essential bound, and taking the supremum over finite subsets gives a finite $N_s(u(t))$. The two norm-equivalence bounds in [F7] then give a pointwise $H^s$ bound. The same argument applies to $v$. With $K^{\mathrm{\prime }}$ as the compact value set and a common finite radius $R_{\ast }$ for these pointwise bounds, step 1.2 and [F4] give $u=v$ on $(-T,T)$. The continuous-dependence estimate in [F4] is exactly the claimed $H^{s-1}$ statement for the solutions from 1.1.

When the initial state is smooth, [F5] applies to the same solution from 1.1 and preserves smoothness on that same interval. No new interval or coupled Einstein-Maxwell conclusion is inferred.