Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: Shocks and contact discontinuities

Abstract

This work is a continuation of our previous work [Z.Q. Shao, Global structure stability of Riemann solutions for linearly degenerate hyperbolic conservation laws under small BV perturbations of the initial data, Nonlinear Anal. Real World Appl. 11 (2010) 3791–3808]. In the present paper we investigate the global structure stability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws under small BV perturbations of the initial data, where the Riemann solution only contains the shocks and contact discontinuities, and at least a shock wave. The perturbations are in BV but they are assumed to be C 1 -smooth, with bounded and possibly large C 1 -norms. We get a lower bound of the lifespan of the piecewise C 1 solution to a class of the generalized Riemann problem, which can be regarded as a small BV perturbation of the corresponding Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics, particularly to one-dimensional Euler equations of gas dynamics for a compressible, inviscid, non-heat conducting gas in Eulerian coordinates, are also given.