The Riemann problem in gas dynamics

Source abstract

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.