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. . . For some of these methods, general L1 convergence results are available.

In this talk we focus on the local behavior of these approximations, at a 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 -approximate solutions, a general expectation is that, by a suitable local rescaling of coordinates, as → 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 with = 1. Some results in this direction will be given, as proved in [BCS] for the case of vanishing viscosity.

[BCS] A. Bressan, L. Caravenna and W. Shen, Local asymptotic patterns for viscous approximations of conservation laws. To appear.