Existence and uniqueness theorems for Riemann problems

Abstract:

In [2] the author proposed the entropy condition (E) and solved the Riemann problem for general $2 \times 2$ conservation laws ${u_t} + f{(u,v)_x} = 0,{v_t} + g{(u,v)_x} = 0$, under the assumptions that the system is hyperbolic, and ${f_u} \geqslant 0$ and ${g_v} \leqslant 0$. The purpose of this paper is to extend the above results to a much wider class of $2 \times 2$ conservation laws. Instead of assuming that ${f_u} \geqslant 0$ and ${g_v} \leqslant 0$, we assume that the characteristic speed is not equal to the shock speed of different family. This assumption is motivated by the works of Lax [1] and Smoller [4].