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The aim of these notes is to provide a self-contained presentation of recent results on hyperbolic systems of conservation laws, based on the vanishing viscosity approach.

Partial differential equations are often approximated by finite difference schemes. The consistency and stability of a given scheme are usually studied through a linearization along elementary solutions, for instance constants. So long as time-dependent problems are concerned, one may also ask for the behaviour of schemes about traveling waves. A rather complete study was made by Chow & al. [12] in the context of fronts in reaction-diffusion equations, for instance KPP equation; see also the monograph by Fiedler & Scheurle [17] for different aspects of the same problem. We address here similar questions in the context of hyperbolic systems of conservation laws. Besides constants, traveling waves may be either linear waves, corresponding to a linear characteristic field, or simple discontinuities such as shock waves of various kinds: Lax shocks, under-compressive shocks, overcompressive ones, anti-Lax ones.

In the first four lectures we describe a recent proof of the short time existence of curved multidimensional viscous shocks, and the associated justification of the small viscosity limit for piecewise smooth curved inviscid shocks. Our goal has been to provide a detailed, readable, and widely accessible account of the main ideas, while avoiding most of the technical aspects connected with the use of pseudodifferential (or paradifferential) operators. The proof might be described as a combination of ODE/dynamical systems analysis with microlocal analysis, with the main new ideas coming in on the ODE side. In a sense the whole problem can be reduced to the study of certain linear systems of nonautonomous ODEs depending on frequencies as parameters. The frequency-dependent matrices we construct as conjugators or symmetrizers in the process of proving estimates for those ODEs serve as principal symbols of pseudodifferential operators used to prove estimates for the original PDEs.

We present a streamlined account of recent developments in the stability theory for planar viscous shock waves, with an emphasis on applications to physical models with “real,” or partial viscosity. The main result is the establishment of necessary, or “weak”, and sufficient, or “strong”, conditions for nonlinear stability analogous to those established by Majda [M.1, M.2, M.3] in the inviscid case but (generically) separated by a codimension-one set in parameter space rather than an open set as in the inviscid case. The importance of codimension one is that transition between nonlinear stability and instability is thereby determined, lying on the boundary set between the open regions of strong stability and strong instability (the latter defined as failure of weak stability). Strong stability holds always for small-amplitude shocks of classical “Lax” type [PZ, FreS]; for large-amplitude shocks, however, strong instability may occur [ZS, Z.3].