Local Asymptotic Patterns for Viscous Approximations of Conservation Laws

Abstract

Solutions to hyperbolic conservation laws can be approximated in many different ways: by vanishing viscosity, relaxations, discrete or semi-discrete numerical schemes, approximation with a nonlocal flux, etc$\dots$ For some of these methods, general ${\mathbf{L}}^1$ convergence results are available. Aim of this paper is to understand the local behavior of these approximations, in a neighborhood of point where the hyperbolic solution has a singularity. Specifically: a point along a shock, or where two shocks interact, or where a new shock is formed. Given a sequence of $\epsilon$-approximate solutions, a general expectation is that, by a suitable local rescaling of coordinates, as $\epsilon \to 0$ a well defined limit is obtained. This corresponds to a specific ``eternal solution" (globally defined both in space and in time) to the approximating equation. Precise results this direction are here given, in the case of vanishing viscosity.