The Riemann problem in gas dynamics

We consider the Riemann problem (R.P.) for the 3 × 3 3\, \times \, 3 system of gas dynamics equations in a single space variable. We assume that the specific internal energy e = e ( v , s ) e = e(v,\,s) (s = specific entropy, v = specific volume) satisfies the usual hypotheses, p v > 0 , p v v > 0 , p s > 0 ( p = − e v = {p_v}\, > \,0,\,{p_{vv}}\, > \,0,\,{p_s}\, > \,0\,(p\, = \, - \,{e_v}\, = pressure); we also assume some reasonable hypotheses about the asymptotic behavior of e. We call functions e satisfying these hypotheses energy functions Theorem 1. For any initial data ( U l , U r ) ( U l = ( v l , p l , u l ) ({U_l},\,{U_r})\,({U_l}\, = \,({v_l},\,{p_l},\,{u_l}) , U r = ( v r , p r , u r ) {U_r}\, = \,({v_r},\,{p_r},\,{u_r}) , u = flow velocity), the R. P. has a solution. We introduce two conditions: \[ ( (I) ) ∂ ∂ v p ( v , e ) ⩽ p 2 2 e a m p ; ( v , e > 0 ) , \begin {array}{*{20}{c}}\tag {$\text {(I)}$} {\frac {\partial } {{\partial v}}\,p(v,\,e) \leqslant \frac {{{p^2}}} {{2e}}} & {(v,\,e\, > \,0),} \\ \end {array} \] \[ ( (II) ) ∂ ∂ v e ( v , p ) ⩾ − p 2 a m p ; ( v , p > 0 ) . \begin {array}{*{20}{c}}\tag {$\text {(II)}$} {\frac {\partial }{{\partial v}}\,e(v,\,p)\, \geqslant - \frac {p} {2}} & {(v,\,p\, > \,0).} \\ \end {array} \] Theorem 2. (I) is necessary and sufficient for uniqueness of solutions of the R. P. Nonuniqueness persists under small perturbations of the initial data. (I) is implied by the known condition ( ( ∗ ) ) ∂ ∂ v e ( v , p ) > 0 ( v , p > 0 ) , \begin{equation}\tag {$(\ast )$} {\frac {\partial } {{\partial v}}e(v,p) > 0} \qquad (v,p > 0), \end{equation} which holds for all usual gases. (I) implies (II). We construct energy functions e that violate (II), that satisfy (II) but violate (I), and that satisfy (I) but violate (*). In all solutions considered, the shocks satisfy the entropy condition and the Lax shock conditions.