Uniqueness of classical solutions and continuous dependence in H^{s-1} for periodic symmetric hyperbolic systems

(Homogeneous forcing and constant dependence.) In (a) the common solution norm bound is $R$; in (b) put $M=C_{\mathrm{e}\mathrm{x}}(1+R)$, where $C_{\mathrm{e}\mathrm{x}}$ is the bound furnished by Local existence of Sobolev and classical solutions of a periodic symmetric hyperbolic system, uniform on bounded H^s data, and use $M$ as the norm radius in [L3], [L4]. Thus the comparison constants in (b) depend on $A^{\mu },B,K,K^{\mathrm{\prime }},R,s$ and the fixed coefficient time slab, through $C_{\mathrm{e}\mathrm{x}}$; no independence from $K$ is inferred. Put $r=s-1$ and $w=u-v$, and define $f=-(A^0(u)-A^0(v)){\mathrm{\partial }}_{t}v-(A^i(u)-A^i(v)){\mathrm{\partial }}_{i}v-B(u)+B(v)$. The difference bound of [L3] and the $H^r$ algebra bound give $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^r}\le C\mathrm{\parallel }w{\mathrm{\parallel }}_{H^r}$, since the derivatives of $v$ have bounded $H^r$ norms by [L4]. Smooth cutoff extension in the proved supplier gives these estimates on the actual compact value sets without assuming convexity. Total coefficient derivatives $\dot{a}=A_t^0(u)+A_u^0(u){\mathrm{\partial }}_{t}u$ and ${\mathrm{\partial }}_i{a}_i$ are bounded in $L^{\mathrm{\infty }}$ by those same solution bounds.

($L^2$ uniqueness for all stated classical solutions.) Set $E_0(t)=\int w^{\mathsf{T}}aw$. Symmetry and integration by parts give $E_0^{\mathrm{\prime }}=\int w^{\mathsf{T}}(\dot{a}+{\mathrm{\partial }}_i{a}_i)w+2\int w^{\mathsf{T}}f$. Pointwise coefficient Lipschitz bounds give $\mathrm{\mid }f\mathrm{\mid }\le C\mathrm{\mid }w\mathrm{\mid }$, since ${\mathrm{\partial }}_{t}v$ and ${\mathrm{\partial }}_{i}v$ are bounded in $L^{\mathrm{\infty }}$. Thus $\mathrm{\mid }{E}_0^{\mathrm{\prime }}\mathrm{\mid }\le C{E}_0$. If $w(0)=0$, Gronwall on either oriented segment from $0$ to $t$ gives $E_0(t)=0$. As $t\in J$ was arbitrary, this proves (a).

(Energy identity at the stated regularity.) Work on any compact time segment and define $E={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r}\int (D^{\alpha }w{)}^{\mathsf{T}}a{D}^{\alpha }w$. Put $w_N=P_{N}w$, $h={\mathrm{\partial }}_{t}w$, and $y_{\alpha }=D^{\alpha }w$, $h_{\alpha }=D^{\alpha }h$. Classical $C^1$ regularity makes each Fourier coefficient $C^1$ with derivative the coefficient of $h$, so $w_N$ is a $C^1$ finite-dimensional curve. The bounds $w\in H^{r+1}$ and $h\in H^r$ are uniform at every time: the former is assumed and the latter follows pointwise from the PDE and [L4]. The coefficients $a$ and $\dot{a}$ are continuous, with $a$ uniformly $C^1$ in space and $\dot{a}$ uniformly bounded. Let $E_N={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r}\int (P_N{y}_{\alpha }{)}^{\mathsf{T}}a(P_N{y}_{\alpha })$. The tail bound [L6] gives uniform $L^2$ convergence $P_N{y}_{\alpha }\to y_{\alpha }$ at rate $O((1+N^2{)}^{-1\mathrm{/}2})$ since $y_{\alpha }\in H^1$ uniformly. Hence $E_N\to E$ uniformly. Finite-dimensional differentiation gives $E_N^{\mathrm{\prime }}={\sum }_{\alpha }[2⟨P_N{h}_{\alpha },a{P}_N{y}_{\alpha }⟩+⟨P_N{y}_{\alpha },\dot{a}{P}_N{y}_{\alpha }⟩]$. By self-adjointness the first pairing equals $⟨h_{\alpha },P_N(a{P}_N{y}_{\alpha })⟩$. Now $a{y}_{\alpha }$ has uniformly bounded $H^1$ norm by the Leibniz rule, and $\mathrm{\parallel }{P}_N(a{P}_N{y}_{\alpha })-a{y}_{\alpha }{\mathrm{\parallel }}_2\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{P}_N{y}_{\alpha }-y_{\alpha }{\mathrm{\parallel }}_2+\mathrm{\parallel }(P_N-I)(a{y}_{\alpha }){\mathrm{\parallel }}_2\to 0$ uniformly by [L6]. The quadratic terms also converge uniformly. Thus $E_N^{\mathrm{\prime }}$ converges uniformly to $G={\sum }_{\alpha }[2⟨h_{\alpha },a{y}_{\alpha }⟩+⟨y_{\alpha },\dot{a}{y}_{\alpha }⟩]$. Integrating the scalar identities $E_N(t)-E_N(t^{\mathrm{\prime }})={\int }_{t^{\mathrm{\prime }}}^t{E}_N^{\mathrm{\prime }}$ and passing uniformly to the limit gives $E(t)-E(t^{\mathrm{\prime }})={\int }_{t^{\mathrm{\prime }}}^{t}G$; $G$ is continuous as a uniform limit of continuous functions. Therefore $E^{\mathrm{\prime }}=G$. The differentiated difference equation is an $L^2$ identity because $w\in H^{r+1}$ and $h\in H^r$; spatial integration by parts is justified by Fourier density and the $H^1$ bounds for $y_{\alpha }$. It may therefore be substituted into $G$ without additional time regularity assumptions.

(Homogeneous difference time derivative.) Invert $a$ in [L1]. Multiplier bounds at order $r-1$, the forcing estimate of step 1.1 and $\mathrm{\parallel }{\mathrm{\partial }}_{i}w{\mathrm{\parallel }}_{H^{r-1}}\le \mathrm{\parallel }w{\mathrm{\parallel }}_{H^r}$ give $\mathrm{\parallel }{\mathrm{\partial }}_{t}w{\mathrm{\parallel }}_{H^{r-1}}\le C\mathrm{\parallel }w{\mathrm{\parallel }}_{H^r}$.

(Order-$r$ difference energy.) Put $E={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r}\int (D^{\alpha }w{)}^{\mathsf{T}}a{D}^{\alpha }w$, so $a_0\mathrm{\parallel }w{\mathrm{\parallel }}_{H^r}^2\le E\le b_0\mathrm{\parallel }w{\mathrm{\parallel }}_{H^r}^2$. Differentiating [L1] gives $a{\mathrm{\partial }}_t{D}^{\alpha }w+a_i{\mathrm{\partial }}_i{D}^{\alpha }w=D^{\alpha }f-[D^{\alpha },a]{\mathrm{\partial }}_{t}w-[D^{\alpha },a_i]{\mathrm{\partial }}_{i}w$. After integration by parts the energy derivative consists of the $\dot{a}+{\mathrm{\partial }}_i{a}_i$ quadratic terms and twice the pairings with these three right-side terms. Step 1.1 bounds the quadratic and forcing terms by $C\mathrm{\parallel }w{\mathrm{\parallel }}_{H^r}^2$. Homogeneous commutators from [L3], together with step 2.1, bound the time commutators by the same expression. Spatial commutators also have that bound since $\mathrm{\parallel }{\mathrm{\partial }}_{i}w{\mathrm{\parallel }}_{H^{r-1}}\le \mathrm{\parallel }w{\mathrm{\parallel }}_{H^r}$. Therefore $\mathrm{\mid }{E}^{\mathrm{\prime }}\mathrm{\mid }\le CE$. The energy identity is justified at this regularity by step 1.3.

(Continuous dependence.) Gronwall in both directions gives $E(t)\le E(0)e^{C\mathrm{\mid }t\mathrm{\mid }}$. Equivalence yields $\mathrm{\parallel }w(t){\mathrm{\parallel }}_{H^r}\le \sqrt{b_0\mathrm{/}{a}_0}{e}^{C\mathrm{\mid }t\mathrm{\mid }\mathrm{/}2}\mathrm{\parallel }w(0){\mathrm{\parallel }}_{H^r}$. Choose $C^{\mathrm{\prime }}$ to dominate $\sqrt{b_0\mathrm{/}{a}_0}$ and $C\mathrm{/}2$. This proves (b).