Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities

Abstract

In this paper, the author proves the global structure stability of the Lax's Riemann solution u=U( x t ) , containing only shocks and contact discontinuities, of general n× n quasilinear hyperbolic system of conservation laws. More precisely, the author proves the global existence and uniqueness of the piecewise C 1 solution u= u( t, x) of a class of generalized Riemann problem, which can be regarded as a perturbation of the corresponding Riemann problem, for the quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to that of the solution u=U( x t ) . Combining the results in Kong (Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves, to appear), the author proves that the Lax's Riemann solution of general n× n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.