Local well-posedness and finite-speed independence of the extension for symmetric hyperbolic systems on a coordinate neighbourhood

(Local existence (a).) By [L1] the solution $u$ is a classical $C^1$ solution on all of $(-T,T)\times {\mathbb{T}}_{2\pi }^3$, in particular near $(0,x_0)$.

(Quasilinear independence (b).) Put $w=u-v$. In the common chart the coefficient functions are the same functions of $(t,x,z)$, but their evaluations at $u$ and $v$ need not agree. Subtraction gives ${\tilde{A}}^{\mu }(t,x,u){\mathrm{\partial }}_{\mu }w+\tilde{C}(t,x)w=0$, where $\tilde{C}w=({\tilde{A}}^{\mu }(u)-{\tilde{A}}^{\mu }(v)){\mathrm{\partial }}_{\mu }v+\tilde{B}(u)-\tilde{B}(v)$. On the compact cone both ranges form a compact subset of the common open state set. A smooth cutoff in the state variable extends each coefficient off that compact neighborhood; the explicit cutoff construction of Smooth composition, difference, coefficient product and commutator estimates on bounded H^s data then gives the mean-value expression for $\tilde{C}$ on the straight segment between $u,v$, without assuming the original state set convex. The coefficient derivatives and the first derivatives of $v$ are bounded there, so $\tilde{C}$ is continuous and bounded. The evaluated leading matrices are $C^1$, symmetric and positive. For this regularity the flux calculation is direct: $j^{\mu }=w^{\mathsf{T}}{\tilde{A}}^{\mu }(u)w$ is $C^1$ and ${\mathrm{\partial }}_{\mu }{j}^{\mu }=w^{\mathsf{T}}({\mathrm{\partial }}_{\mu }{\tilde{A}}^{\mu }-\tilde{C}-{\tilde{C}}^{\mathsf{T}})w\le c{j}^0$. To check the actual cone boundary, write $\varphi (t,y)=d(x(y),x_0)+\lambda (t-t_0)$. At each smooth lateral boundary point before the vertex its outward conormal is a positive multiple of $(\lambda ,{\xi }_i)$ with ${\xi }_i={\mathrm{\partial }}_{y^i}d(x(y),x_0)$ and $h^{ij}{\xi }_i{\xi }_j=1$: locally the distance is $\mathrm{\mid }x(y)-x_0+2\pi k\mathrm{\mid }$ for a nearest lattice translate, whose Euclidean gradient has norm one. Thus $j^{\mu }{\mathrm{\partial }}_{\mu }\varphi =w^{\mathsf{T}}(\lambda {\tilde{A}}^0+{\xi }_i{\tilde{A}}^i)w\ge 0$ by the stated flat-dual-norm condition, in this very chart. On compact subsets only finitely many lattice translates can be nearest. Partitioning into their nearest-point regions gives piecewise smooth cone faces; internal partition fluxes cancel, and intersections of faces have surface measure zero. A finite local lifting cover gives the same flux calculation if the chart has no single lift. For truncated subcones $\tau <t_0$ the vertex is excluded, so the divergence theorem applies to these finitely partitioned domains. A speed bound exists because $A^0\ge aI$ for some $a>0$ and $b:=\sup \mathrm{\parallel }{\xi }_i{A}^i\mathrm{\parallel }<\mathrm{\infty }$ over the compact coefficient set and its flat-unit covectors; ${\lambda }_0=b\mathrm{/}a+1$ suffices. Rescaling a chart changes both the coefficient components and the dual-metric unit condition, preserving this inequality. The divergence theorem and Gronwall give $E(\tau )\le E(0)+c{\int }_0^{\tau }E$ and $E(0)=0$, hence $w=0$ on the cone interior and, by continuity, also its boundary and vertex. This is the energy argument of [L3], with its needed regularity established rather than asserting identical composed coefficients.

((c).) The assertions are exactly [L4] applied to the solutions under consideration.