Local Asymptotic Patterns for Viscous Approximations of Conservation Laws

TL;DR

This paper investigates the local behavior of viscous approximations to hyperbolic conservation laws near singularities, revealing that rescaled solutions converge to specific eternal solutions as viscosity vanishes.

Contribution

It provides rigorous results on the local asymptotic patterns of viscous approximations near shocks and interactions, enhancing understanding of their singularity structure.

Findings

Rescaled solutions near singularities converge to eternal solutions.

Analysis applies specifically to vanishing viscosity approximations.

Results clarify local behavior around shocks and shock interactions.

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$\ldots$ For some of these methods, general ${\bf L}^1$ 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.