Stability of surfaces of strong discontinuity in gases

Abstract

The existence of solutions with surfaces of strong discontinuity is one of the principal features of the continua whose motions are described by systems of differential equations of hyperbolic type. Shock waves in gas dynamics, magnetohydrodynamics and in solids, detonation waves and combustion fronts, contact discontinuities, etc. are well-known examples of these surfaces. The discontinuities are usually investigated in accordance with the following scheme: 1) derivation of the boundary conditions on the discontinuity from the input system of differential equations in integral form; 2) verification of the fulfilment of the evolution conditions; 3) solution of the problem of the discontinuity structure and, when the occasion requires, obtaining supplementary boundary conditions; 4) investigation of the stability of the discontinuity. Only after obtaining positive results in all fours stages can we assert that the existence of the discontinuity is theoretically justified and that it can be used for constructing the solutions of particular boundary value problems. In the present paper attention will be concentrated on the problem of the stability of discontinuities, all the material, with the exception of the general results of Sec.1, being concerned with gas media and relating to discontinuities on whose surface the normal mass flow is nonzero. Having no way of exploring all the aspects of the problem of the stability of discontinuities in the same detail within the limited context of this paper, the authors hope to demonstrate the most general ideas and approaches which could subsequently be used to investigate the stability of discontinuities in various particular models of continua.