摘要：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 $L^1$ convergence results are available.  The talk will outline a general approach to understand the local asymptotic behavior of these approximations, at a point where the hyperbolic solution has a singularity. Specifically: points along a shock, or where two shocks interact, or points where a new shock is formed. This leads to a wide array of open problems. Given a sequence of $\epsilon$-approximate solutions, by a suitable local rescaling of coordinates we expect that as $\epsilon\to 0$, a well defined limit can be obtained. This corresponds to a globally defined (both in space and time) solution to the approximating equation with $\epsilon = 1$. 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.