L^1 stability of patterns of non-interacting large shock waves

Abstract

We consider a strictly hyperbolic n × n system of conservation laws in one space dimension ut + f(u)x = 0, together with Cauchy initial data u(0, x) = ū(x), that is a small BV ∩ L perturbation of fixed Riemann data (u− 0 , u 0 ). We a priori assume that the latter problem is solved by M large shocks (2 ≤ M ≤ n) of different characteristic families, each of them Majda stable and Lax compressive. We prove that under a suitable Finiteness Condition the problem has a unique solution defined globally in space and time, while a stronger Stability Condition guarantees the existence of a Lipschitz semigroup of solutions.