Complete symmetrized constant-coefficient evolution

For any real symmetric matrix $H$, maximize $z^{\mathsf{T}}Hz$ on the compact unit sphere. Differentiating along every tangent direction shows the maximizing vector is an eigenvector; its orthogonal complement is invariant under $H$, so induction gives an orthonormal eigenbasis. Applying this to $M$ constructs $M^{1\mathrm{/}2}$ and $M^{-1\mathrm{/}2}$. The condition $M{A}_j=A_j^{\mathsf{T}}M$ makes ${\tilde{A}}_j=M^{1\mathrm{/}2}{A}_j{M}^{-1\mathrm{/}2}$ symmetric. Therefore each eigenvalue in $v_{\ast }$ is real and has absolute value at most ${\sum }_j\mathrm{\parallel }{\tilde{A}}_j\mathrm{\parallel }$, so $v_{\ast }<\mathrm{\infty }$. With $e=U^{\mathsf{T}}MU\mathrm{/}2$ and $F_j=U^{\mathsf{T}}M{A}_{j}U\mathrm{/}2$, the product rule gives ${\mathrm{\partial }}_{t}e+{\sum }_j{\mathrm{\partial }}_{x_j}{F}_j=-U^{\mathsf{T}}SU\le 0$ for every smooth solution, and $\mathrm{\mid }F\cdot \nu \mathrm{\mid }\le v_{\ast }e$ by diagonalizing $\tilde{A}(\nu )$.

For Schwartz data, [F1] permits componentwise Fourier transformation of the system, giving ${\mathrm{\partial }}_t\widehat{U}=C(\xi )\widehat{U}$ with $C(\xi )=-2\pi i{\sum }_j{\xi }_j{A}_j-B$. The matrix exponential $E(t,\xi )=e^{tC(\xi )}$, defined by its locally uniformly convergent power series, solves this finite-dimensional ODE. For $z(t)=E(t,\xi )z_0$, $\frac{d}{dt}(z^{\ast }Mz)=-2z^{\ast }Sz\le 0$, because $MA(\xi )$ is real symmetric and its multiple by $-i$ has zero real quadratic part. Thus $\mathrm{\parallel }E(t,\xi ){\mathrm{\parallel }}_M\le 1$ for $t\ge 0$, uniformly in $\xi$; for $t<0$, $\mathrm{\mid }\frac{d}{dt}(z^{\ast }Mz)\mathrm{\mid }\le 2\mathrm{\parallel }{M}^{-1\mathrm{/}2}S{M}^{-1\mathrm{/}2}\mathrm{\parallel }{z}^{\ast }Mz$, so Grönwall gives a uniform $e^{C\mathrm{\mid }t\mathrm{\mid }}$ bound on bounded time intervals.

Differentiating the matrix exponential with respect to ${\xi }_j$ gives ${\mathrm{\partial }}_{{\xi }_j}E(t,\xi )={\int }_0^{t}E(t-s,\xi )(-2\pi i{A}_j)E(s,\xi )ds$. Repeating this identity expresses every fixed $\xi$-derivative as a finite sum of ordered insertion integrals with constant matrix factors between evolution matrices. The uniform bounds from step 1.2 and the finite integration-simplex volumes bound each such derivative uniformly in $\xi$ on compact time intervals. Time derivatives multiply by $C(\xi )$ and therefore have at most polynomial growth in $\xi$. Leibniz's rule now shows $E(t,\cdot ){\widehat{U}}_0$ is Schwartz, uniformly in its seminorms on compact time intervals.

For periodic smooth data, integration by parts in a coordinate with maximal $\mathrm{\mid }{k}_j\mathrm{\mid }$ gives $\mathrm{\mid }{\widehat{U}}_k(0)\mathrm{\mid }\le C_N(1+\mathrm{\mid }k\mathrm{\mid }{)}^{-N}$ for every $N$. The series $U(t,x)={\sum }_{k\in {\mathbb{Z}}^3}{e}^{2\pi ik\cdot x\mathrm{/}L}E(t,k\mathrm{/}L){\widehat{U}}_k(0)$ and every differentiated series converge uniformly on compact time intervals, by the same matrix-exponential derivative bounds and the lattice-shell estimate that a unit-width shell in ${\mathbb{Z}}^3$ has $O((1+r{)}^2)$ points. Hence it is a global smooth periodic solution and solves the PDE termwise. The modewise $M$-estimate from step 1.2 yields contraction of every weighted coefficient norm. For $s>3\mathrm{/}2$, Cauchy–Schwarz and $M\ge {\lambda }_{\min }(M)I$ give ${\sum }_k\mathrm{\mid }{\widehat{U}}_k(t)\mathrm{\mid }\le {\lambda }_{\min }(M{)}^{-1\mathrm{/}2}({\sum }_k(1+\mathrm{\mid }k\mathrm{/}L{\mathrm{\mid }}^2{)}^{-s}{)}^{1\mathrm{/}2}({\sum }_k(1+\mathrm{\mid }k\mathrm{/}L{\mathrm{\mid }}^2{)}^s{\widehat{U}}_k(t{)}^{\ast }M{\widehat{U}}_k(t){)}^{1\mathrm{/}2}$; the first sum converges by the shell bound, proving the asserted uniform norm estimate. Integrating the equation against each periodic exponential shows that every solution coefficient solves the same finite-dimensional ODE, proving uniqueness.

Define $U(t)={\mathcal{F}}^{-1}(E(t,\cdot ){\widehat{U}}_0)$. By [F2] this inverse is pointwise the absolutely convergent Fourier integral; the decay proved in step 2.1 permits every spatial and time derivative to pass under that integral, proves the PDE, and recovers $U(0)=U_0$. Conjugation symmetry of $C(\xi )$ and ${\widehat{U}}_0(-\xi )=\overline{{\widehat{U}}_0(\xi )}$ makes $U$ real.

By [F3] the Plancherel transform is unitary, and [F4] identifies it with the integral transform in ${\mathcal{S}}^{\mathrm{\prime }}$. Multiplying the pointwise $M$-norm estimate in step 1.2 by $(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^s$ and integrating gives the stated $H_M^s$ contraction for every $s\ge 0$.

Fix $t_{\ast }>0$, $x_{\ast }$, and $0\le t<t_{\ast }$. Put $R(t)=v_{\ast }(t_{\ast }-t)$ and $E(t)={\int }_{B_{R(t)}(x_{\ast })}e(t,x)dx$. In polar coordinates, the fundamental theorem of calculus differentiates this moving-ball integral; the PDE energy identity and [F5] then give $E^{\mathrm{\prime }}(t)=-{\int }_{\mathrm{\partial }{B}_{R(t)}}(F\cdot \nu +v_{\ast }e)dS-{\int }_{B_{R(t)}}{U}^{\mathsf{T}}SUdx\le 0$, since $\mathrm{\mid }F\cdot \nu \mathrm{\mid }\le v_{\ast }e$. If the initial data vanish on $B_{v_{\ast }{t}_{\ast }}(x_{\ast })$, then $E(0)=0$; positivity of $M$ and $E(t)\ge 0$ force $U(t,x_{\ast })=0$ by continuity as $t\uparrow t_{\ast }$. For any $x$ with $\mathrm{\mid }x-x_0\mathrm{\mid }<R-v_{\ast }{t}_{\ast }$, the ball $B_{v_{\ast }{t}_{\ast }}(x)$ is contained in $B_R(x_0)$, proving the stated shrinking-ball conclusion. Applying the same argument to the difference of two solutions proves local uniqueness. Periodic lifts give the identical geodesic-ball conclusion on ${\mathbb{T}}_L^3$. If $v_{\ast }=0$, every symmetric ${\tilde{A}}_j$ has zero spectrum and hence is zero, so each $A_j=0$; the pointwise ODE preserves spatial support.