Stability of $L^{\infty }$ solutions for hyperbolic systems with coinciding shocks and rarefactions

Abstract

We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction curves and the existence of a set of Riemann coordinates $w$, we prove that there exists a semigroup of solutions $u(t)={\mathcal{S}}_t{u}_0$, defined on initial data $u_0\in L^{\infty }$. The semigroup $\mathcal{S}$ is continuous w.r.t. time and the initial data $u_0$ in the $L_{\text{loc}}^1$ topology. Moreover $\mathcal{S}$ is unique and its trajectories are obtained as limits of wave front tracking approximations.

19 pages, 13 figures