The Riemann problem for general $2\times 2$ conservation laws

Abstract:

The Riemann Problem for a system of hyperbolic conservation laws of form \[ (1)\quad \begin {array}{*{20}{c}} {{u_t} + f{{(u,\upsilon )}_x} = 0,} \\ {{\upsilon _t} + g{{(u,\upsilon )}_x} = 0} \\ \end {array} \] with arbitrary initial constant states \[ (2)\quad ({u_0}(x),{v_0}(x)) = \left \{ {\begin {array}{*{20}{c}} {({u_l},{v_l}),\quad x < 0,} \\ {({u_r},{v_r}),\quad x > 0,} \\ \end {array} } \right .\] is considered. We assume that ${f_\upsilon } < 0,{g_u} < 0$. Let ${l_i}({r_i})$ be the left (right) eigenvectors of $dF \equiv d(f,g)$ for eigenvalues ${\lambda _1} < {\lambda _2}$. Instead of assuming the usual convexity condition $d{\lambda _i}({r_i}) \ne 0,i = 1,2$ we assume that $d{\lambda _i}({r_i}) = 0$ on disjoint union of $1$-dim manifolds in the $(u,\upsilon )$ plane. Oleinik’s condition (E) for single equation is extended to system (1); again call this new condition (E). Our condition (E) implies Lax’s shock inequalities and, in case $d{\lambda _i}({r_i}) \ne 0$, the two are equivalent. We then prove that there exists a unique solution to the Riemann Problem (1) and (2) in the class of shocks, rarefaction waves and contact discontinuities which satisfies condition (E).