Strict hyperbolicity and Lax shock pattern

For a one-dimensional system $U_t+{\mathrm{\partial }}_{x}F(U)=0$ with $F\in C^1(\mathcal{O};{\mathbb{R}}^m)$ on an open state domain $\mathcal{O}\subseteq {\mathbb{R}}^m$, the system is strictly hyperbolic at $V\in \mathcal{O}$ if the Jacobian $DF(V)$ has $m$ distinct real eigenvalues. Write them in increasing order as ${\lambda }_1(V)<\cdots <{\lambda }_m(V)$. For states $U_L,U_R\in \mathcal{O}$ and speed $s\in \mathbb{R}$ satisfying the Rankine–Hugoniot equality $F(U_R)-F(U_L)=s(U_R-U_L),$ assume strict hyperbolicity at both states. The jump is a $k$-Lax shock, $1\le k\le m$, if ${\lambda }_k(U_R)<s<{\lambda }_k(U_L),$ and also ${\lambda }_{k-1}(U_L)<s$ when $k>1$, and $s<{\lambda }_{k+1}(U_R)$ when $k<m$. The outer inequality is omitted at each endpoint $k=1$ or $k=m$.

one-dimensional conservation system

These inequalities describe the incoming-characteristic pattern at a shock. They are a definition: strict hyperbolicity and the Rankine–Hugoniot equality alone do not imply them. This definition makes no claim that a system has such a shock, or that a Riemann or Cauchy problem has a solution or a unique solution.