Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws under small BV perturbations of the initial data: shocks and contact discontinuities

This work is a continuation of our previous work (Shao, Nonlinear Anal. Real World Appl. 11 (2010) 3791–3808) ‘Global structure stability of Riemann solutions for linearly degenerate hyperbolic conservation laws under small BV perturbations of the initial data’. In the present paper, we prove the global structure instability of the Lax's Riemann solution u=U(x/t) containing only shocks and contact discontinuities, and at least a shock wave, of general n×n quasilinear hyperbolic systems of conservation laws under small BV perturbations of the Riemann initial data. The perturbations are in BV but they are assumed to be C¹ smooth, with bounded and possibly large C¹ norms. We prove the non-existence of global piecewise C¹ solution to a class of the generalized Riemann problem, which can be regarded as a small BV perturbation of the corresponding Riemann problem, for general n×n quasilinear hyperbolic systems of conservation laws. 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.