Local asymptotic patterns for viscous approximations of conservation laws

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. The aim of this paper is to understand the local behavior of these approximations, in a neighborhood of 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 ε-approximate solutions, a general expectation is that, by a suitable local rescaling of coordinates, as ε→ 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 in this direction are here given, in the case of vanishing viscosity.