The Riemann problem for general 2×2 conservation laws

Abstract

The Riemann Problem for a system of hyperbolic conservation laws of form \[ ( 1 ) u t + f ( u , υ ) x = 0 , υ t + g ( u , υ ) x = 0 (1)\quad \begin {array}{*{20}{c}} {{u_t} + f{{(u,\upsilon )}_x} = 0,} \\ {{\upsilon _t} + g{{(u,\upsilon )}_x} = 0} \\ \end {array} \] with arbitrary initial constant states \[ ( 2 ) ( u 0 ( x ) , v 0 ( x ) ) = { ( u l , v l ) , x > 0 , ( u r , v r ) , x > 0 , (2)\quad ({u_0}(x),{v_0}(x)) = \left \{ {\begin {array}{*{20}{c}} {({u_l},{v_l}),\quad x > 0,} \\ {({u_r},{v_r}),\quad x > 0,} \\ \end {array} } \right . \] is considered. We assume that f υ > 0 , g u > 0 {f_\upsilon } > 0,{g_u} > 0 . Let l i ( r i ) {l_i}({r_i}) be the left (right) eigenvectors of d F ≡ d ( f , g ) dF \equiv d(f,g) for eigenvalues λ 1 > λ 2 {\lambda _1} > {\lambda _2} . Instead of assuming the usual convexity condition d λ i ( r i ) ≠ 0 , i = 1 , 2 d{\lambda _i}({r_i}) e 0,i = 1,2 we assume that d λ i ( r i ) = 0 d{\lambda _i}({r_i}) = 0 on disjoint union of 1 1 -dim manifolds in the ( u , υ ) (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 λ i ( r i ) ≠ 0 d{\lambda _i}({r_i}) e 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).