Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws

Abstract

We deal with the non characteristic initial and boundary value problem for an n × n strictly hyperbolic system of conservation laws in one space dimension $$\partial_tu + \partial_xF(u) = 0,\quad u(0, x) = \bar{u}(x),\quad b\left( u(\psi(t), t)\right) = g(t).\quad (*)$$ Here F is a smooth vector field defined in an open, convex neighborhood of the origin of \({\mathbb{R}}^n, \bar{u}\) and g are functions with small total variation, \(x = \psi(t)\) is a non characteristic Lipschitz boundary profile, and b a \({\mathcal{C}}^1\) function. We prove that the front tracking solutions to (*) constructed by D. Amadori in [1] are stable for the \({\mathbb{L}}^1\) topology. This implies the existence of a Standard Riemann Semigroup and hence the well-posedness of (*).