Instability of rarefaction shocks in systems of conservation laws

Abstract

It is well-known that rarefaction shocks are unstable solutions of nonlinear hyperbolic conservation laws. Indeed, for scalar equations rarefaction shocks are unstable in the class of smooth solutions, but for systems one can only say in general that rarefaction shocks are unstable in the larger class of weak solutions. (Here unstable refers to a lack of continuous dependence upon perturbations of the initial data.) Since stability in the class of weak solutions is not well understood, ([T, TE]), “entropy” considerations have played a leading role in ruling out shocks that violate the laws of physics. However, for non-strictly hyperbolic systems the analogy with the equations of gas dynamics breaks down, and general entropy or admissibility criteria for the variety of shocks which appear, (see, e.g., [IMPT]), are not known. In this paper we address the question of when the instability of a shock can be demonstrated within the class of smooth solutions alone. We show by elementary constructions that this occurs whenever there exists an alternative solution to the Riemann problem with the same shock data which consists entirely of rarefaction waves and contact discontinuities with at least one non-zero rarefaction wave. We show that for 2×2 strictly hyperbolic, genuinely nonlinear systems the condition is both necessary and sufficient. We show too that for the full 3×3 (Euler) equations of gas dynamics with polytropic equations of state, rarefaction shocks of “moderate” strength are unstable in the class of smooth solutions if and only if the adiabatic gas constant γ satisfies 1 < γ < 5/3 (see Theorem 8). More precisely, there is a constant y _*, 0 < y _* < 1, depending only on γ, such that if y _* p _l≦p _r≦p _l for 1-shocks, and if y _* p _r≦P _l≦p _r for 3-shocks (where p _r and p _l denote the pressures on both sides of the rarefaction shock), then the shock is unstable if and only if 1 < γ < 5/3. Thus for such shocks, the theory of the Riemann problem for polytropic gases in the range 1 < γ < 5/3 can be rigorously developed with a knowledge of the smooth solutions alone by using stability under smoothing as an admissibility criterion, rather than by using the classical entropy inequalities.