L1 Continuous Dependence Property¶for Systems of Conservation Laws

Abstract:

This paper is concerned with the uniqueness and L ¹ continuous dependence of entropy solutions for nonlinear hyperbolic systems of conservation laws.

We study first a class of linear hyperbolic systems with discontinuous coefficients: Each propagating shock wave may be a Lax shock, or a slow or fast undercompressive shock, or else a rarefaction shock. We establish a result of L ¹ continuous dependence upon initial data in the case where the system does not contain rarefaction shocks. In the general case our estimate takes into account the total strength of rarefaction shocks. In the proof, a new time-decreasing, weighted L ¹ functional is obtained via a step-by-step algorithm.

To treat nonlinear systems, we introduce the concept of admissible averaging matrices which are shown to exist for solutions with small amplitude of genuinely nonlinear systems. Interestingly, for many systems of continuum mechanics, they also exist for solutions with arbitrary large amplitude. The key point is that an admissible averaging matrix does not exhibit rarefaction shocks. As a consequence, the L ¹ continuous dependence estimate for linear systems can be extended to nonlinear hyperbolic systems using a wave-front tracking technique.