Strictly hyperbolic conservation systems and characteristic families

Let $m\ge 1$, let $G\subseteq {\mathbb{R}}^m$ be open, and let $f:G\to {\mathbb{R}}^m$ be $C^3$ (Continuously differentiable maps, local inverses, and local diffeomorphisms; the $C^3$ convention is that of $C^k$ Euclidean maps and diffeomorphisms). The one-dimensional conservation system with state domain $G$ is $u_t+{\mathrm{\partial }}_{x}f(u)=0,u(t,x)\in G,$ where $u$ is the conserved state and $f$ its flux. Its flux Jacobian is $A(u)=Df(u)$. The system is strictly hyperbolic on an open set $N\subseteq G$ if, for every $u\in N$, $A(u)$ has $m$ distinct real eigenvalues, written in increasing order ${\lambda }_1(u)<\cdots <{\lambda }_m(u)$. An $i$th characteristic family on $N$ is a $C^2$ right-eigenvector field $r_i:N\to {\mathbb{R}}^m\setminus \{0\}$ with $A(u)r_i(u)={\lambda }_i(u)r_i(u)$. It is genuinely nonlinear on $N$ if $D{\lambda }_i(u)r_i(u)$ is nowhere zero there; after shrinking $N$ its sign is constant. It is linearly degenerate on $N$ if $D{\lambda }_i(u)r_i(u)=0$ throughout $N$. Rescaling $r_i$ by a nonzero constant changes the wave-strength parameter but not the family.

For $Q=(0,\mathrm{\infty })\times \mathbb{R}$, a locally bounded map $u:Q\to G$ is a distributional weak solution with initial data $u_0$ if ${\int }_Q(u\cdot {\phi }_t+f(u)\cdot {\phi }_x)dtdx=0$ for every vector test function $\phi \in C_c^{\mathrm{\infty }}(Q;{\mathbb{R}}^m)$ and $u(t,\cdot )\to u_0$ in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathbb{R};{\mathbb{R}}^m)$ as $t\downarrow 0$. A Riemann initial datum is the step $u_0(x)=u_L$ for $x<0$ and $u_0(x)=u_R$ for $x>0$.

In this weak-solution clause, $u$ is required to be Lebesgue measurable and $f\circ u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(Q;{\mathbb{R}}^m)$, so the test-function integral is defined. Local boundedness gives $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(Q;{\mathbb{R}}^m)$.