Global structure stability of Riemann solutions for linearly degenerate hyperbolic conservation laws under small BV perturbations of the initial data

Abstract

In this paper, we study the global structure stability of the Riemann solution u = U ( x t ) for general n × n quasilinear hyperbolic systems of conservation laws under a small BV perturbation of the Riemann initial data. We prove the global existence and uniqueness of piecewise C 1 solution containing only n contact discontinuities 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 linearly degenerate quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to the one of the self-similar solution u = U ( x t ) to the corresponding Riemann problem. Our result indicates that this kind of Riemann solution u = U ( x t ) mentioned above for general n × n quasilinear hyperbolic systems of conservation laws possesses a global nonlinear structure stability under a small BV perturbation of the Riemann initial data. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics, particularly to the system describing the motion of the relativistic string in Minkowski space R 1 + n , are also given.