The entropy condition and the admissibility of shocks

The author proposed (Trans. Amer. Math. Soc.199 (1974), 89–112) the extended entropy condition (E) and solved the Riemann problem for general 2 × 2 conservation laws. The Riemann problem for 3 × 3 gas dynamics equations was treated by the author (J. Differential Equations18 (1975), 218–231). In this paper we justify condition (E) by the viscosity method in the spirit of Gelfand Uspehi Mat. Nauk14 (1959), 87–158]. We show that a shock satisfies condition (E) if and only if the shock is admissible, that is, it is the limit of progressive wave solutions of the associated viscosity equations. For the “genuinely nonlinear” 2 × 2 conservation laws, Conley and Smoller Comm. Pure Appl. Math.23 (1970), 867–884] proved that a shock satisfies Lax's shock inequalities cf. Comm. Pure Appl. Math.14 (1957), 537–566] if and only if it is admissible. In this paper, we consider systems that are not necessarily genuinely nonlinear.