University of Victoria, Canada
A basic problem in Kinetic theory is the derivation of shock wave solutions for the Boltzmann equation. This problem is apparently of more than theoretical interest since the (smooth) Boltzmann shock profiles are believed to give better approximations for viscous media than, for example, discontinuous Euler profiles, even though the latter appear as the fluid dynamical limit of the Boltzmann solutions.
In this talk I will set up the problem in the special case of the so-called Discrete Velocity Model (DVM) Boltzmann equation where the formulation reduces to an ODE. The analysis of this ODE can be simplified by a pair of dimensional reductions; first via standard invariants from the kinetic theory and second via a bifurcation argument and centre manifold analysis.
The result: Existence of DVM shock profiles for weak shocks (i.e. the case where the shock connects two equilibria which are not too far apart).