Stability of Multidimensional Viscous Shocks

Summary

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.