Smooth data give smooth solutions of a periodic symmetric hyperbolic system on the existence interval

(Projected time derivative at one higher order.) Fix $n\ge 6$ and a smooth Galerkin field $z$ with $\mathrm{\parallel }z{\mathrm{\parallel }}_{H^{n-1}}\le \rho$ and values in $Q$. Set $g={\mathrm{\partial }}_{t}z$. The lower-order time derivative lemma [L5], used at base order $n-1$, gives $\mathrm{\parallel }g{\mathrm{\parallel }}_{H^{n-2}}\le C_{\rho }$. For $\mathrm{\mid }\alpha \mathrm{\mid }\le n-1$ differentiate $P_N(A^0(z)g)=-P_N(A^i(z){\mathrm{\partial }}_{i}z+B(z))$ and pair with $D^{\alpha }g$. Coercivity gives $a\mathrm{\parallel }{D}^{\alpha }g{\mathrm{\parallel }}_2\le \mathrm{\parallel }{D}^{\alpha }(A^i{\mathrm{\partial }}_{i}z+B){\mathrm{\parallel }}_2+\mathrm{\parallel }[D^{\alpha },A^0]g{\mathrm{\parallel }}_2$. In the source, coefficient derivatives of order at most $n-3$ are $L^{\mathrm{\infty }}$ bounded by $\rho$, and the remaining derivative of ${\mathrm{\partial }}_{i}z$ has order at most $n-1$, hence $L^2$ bound $\mathrm{\parallel }z{\mathrm{\parallel }}_{H^n}$. For coefficient derivatives of order $n-2$ or $n-1$, at most one derivative lands on ${\mathrm{\partial }}_{i}z$, which is $L^{\mathrm{\infty }}$ bounded by $H^{n-1}$ since $n\ge 6$; coefficient derivatives have bounded $L^2$ norms. The commutator has coefficient derivatives at most $n-3$ times $L^2$ derivatives of $g$ of order at most $n-2$, or coefficient derivatives of order at least $n-2$ times at most one $L^{\mathrm{\infty }}$ derivative of $g$, controlled by $H^{n-2}$. Thus $\mathrm{\parallel }g{\mathrm{\parallel }}_{H^{n-1}}\le C_{\rho }(1+\mathrm{\parallel }z{\mathrm{\parallel }}_{H^n})$.

(Base approximation bounds.) Let ${\rho }_s$ be the uniform $H^s$ bound for the Galerkin family of [L1]. These approximations exist on the original $(-T,T)$ independently of their higher norms and have values in $Q$. At every higher order $m$, their initial norms are at most $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^m}$ by projection contraction.

(Induction hypothesis.) Fix $M\ge s$ and assume the same cofinal Galerkin family has a uniform $H^M$ bound ${\rho }_M$ on $(-T,T)$.

(Higher energy controlled by the lower norm.) Define $E_n={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le n}\int (D^{\alpha }z{)}^{\mathsf{T}}{A}^0(z)D^{\alpha }z$. The energy identity of [L5] applies at order $n$: its derivation uses only smoothness, symmetry, projection commutation and integration by parts. The errors are commutators with $g$ and ${\mathrm{\partial }}_{i}z$, the $H^n$ composition of $B$, and quadratic terms with ${\dot{A}}^0$ and ${\mathrm{\partial }}_i{A}^i$. By L2, step 1.1 and the lower $H^{n-2}$ bounds, each commutator and the $B$ term has $L^2$ norm at most $C_{\rho }(1+\mathrm{\parallel }z{\mathrm{\parallel }}_{H^n})$. Total coefficient derivatives have bounded $L^{\mathrm{\infty }}$ norms by the lower bounds for $z,g$. Pairing with $D^{\alpha }z$, summing and using $2xy\le x^2+y^2$ gives $\mathrm{\mid }{E}_n^{\mathrm{\prime }}\mathrm{\mid }\le C_{\rho }(1+E_n)$ and $a\mathrm{\parallel }z{\mathrm{\parallel }}_{H^n}^2\le E_n\le b\mathrm{\parallel }z{\mathrm{\parallel }}_{H^n}^2$, uniformly in $N$. This establishes the high-order refinement before applying induction to the limit.

(Gain one derivative.) Apply step 2.1 with $n=M+1\ge 6$ and $\rho ={\rho }_M$. The approximations are smooth ODE solutions, so their order-$(M+1)$ energies exist before any higher-regularity assertion about the limit. Step 2.1 gives $\mathrm{\mid }{E}_{M+1}^{\mathrm{\prime }}\mathrm{\mid }\le C_M(1+E_{M+1})$ uniformly in $N$, with $E_{M+1}(0)\le b\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^{M+1}}^2$. Gronwall gives $1+E_{M+1}(t)\le (1+b\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^{M+1}}^2)e^{C_{M}T}$. Energy equivalence yields a uniform $H^{M+1}$ bound ${\rho }_{M+1}$ on the original whole interval. Step 1.1 also bounds ${\mathrm{\partial }}_t{u}_N$ in $H^M$. This proves the step from $M$ to $M+1$.

(Higher bounds for the same solution.) Induction supplies these approximation bounds at every order. Finite Fourier-sum limits along the original convergent solution subsequence and its convergent time-derivative subsequence give $u\in L^{\mathrm{\infty }}{H}^m$ and ${\mathrm{\partial }}_{t}u\in L^{\mathrm{\infty }}{H}^{m-1}$ at every order, with the recursively obtained bounds. The time $T$ is unchanged. Uniqueness [L6] identifies any alternative construction with this $u$.

(Joint smoothness.) At every integer $q\ge 0$, the uniform $H^q$ derivative bound and coefficientwise fundamental theorem give $\mathrm{\parallel }u(t)-u(t^{\mathrm{\prime }}){\mathrm{\parallel }}_{H^q}\le C_q\mathrm{\mid }t-t^{\mathrm{\prime }}\mathrm{\mid }$: apply the bound first to finite Fourier sums and take their supremum. Thus $u$ is continuous into every $H^q$. The PDE writes ${\mathrm{\partial }}_{t}u=-(A^0(t,x,u){)}^{-1}(A^i(t,x,u){\mathrm{\partial }}_{i}u+B(t,x,u))$; smooth composition and products, using one extra spatial order, make this right side continuous in every $H^q$. Integrating the coefficientwise identity gives $u\in C^1((-T,T);H^q)$. Repeated differentiation of this smooth coefficient formula in time, with the chain and Leibniz rules and arbitrarily high spatial norms, inductively gives $u\in C^k((-T,T);H^q)$ for all $k,q$. Sobolev embedding makes all mixed spacetime derivatives continuous, proving joint smoothness.