Title: Pointwise semigroup methods for stability of viscous shock waves.
Speaker: Professor Kevin Zumbrun
Speaker Info: Indiana University
In this talk, we present a new, dynamical systems approach to stability of viscous shock waves. Considered as solutions of ODE on $L^p$, stationary viscous shock waves are nonhyperbolic rest points, to which standard semigroup methods do not apply. Indeed, it is well known that decay to shock waves occurs at algebraic, rather than exponential rate. For this reason, there have been until now no results on shock stability from this perspective except in the scalar case, which can be reduced to the standard semigroup setting by Sattinger's method of weighted norms. We overcome this difficulty by the introduction of new, pointwise semigroup methods, by which one can do "hard" analysis in PDE within the dynamical systems framework. Using our new approach, we can treat general over and undercompressive, and even strong shock waves for systems within the same framework used for standard weak (i.e. slowly varying) Lax waves. We can also treat physically realistic viscosity matrices. Save for the standard Lax case with artificial identity viscosity matrix, all of the above-mentioned results are new. The approach should have applications to other situations of sensitive stability as well, for example dispersive undercompressive shocks, combustion waves, and multidimensional shock fronts.Date: Thursday, January 8, 1998