We consider an elliptic system with variational structure. After making some general remarks, we focus on potentials that possess several global minima and are invariant under a finite reflection group G. We establish existence of G-equivariant entire solutions connecting the global minima.

We present recent results with B. Texier developing a rigorous nonlinear theory of stability and bifurcation of strong detonation waves of the full reacting Navier-Stokes (rNS) equations, based on natural spectral stability/bifurcation conditions. We discuss in parallel recent singular perturbation results showing that in the small viscosity limit these conditions reduce to the corresponding conditions for the ZND, or reacting Euler, equations that are more commonly studied in the detonation literature. This yields immediately numerical verification of the (rNS) conditions through the voluminous numerical literature on (ZND).

In this paper, we consider the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations with initial data in the critical Besov-Sobolev type spaces. This is partially joint work with J. Y. Chemin, Guilong GUI and M. Paicu.

I shall talk about some regularity results for the ultraparabolic equation, in particular, the C^{alpha}$ regularity of weak solutions. The problem arises from the Prandtl's boundary layer system under the Crocco transformation. I shall also report some recent results on the backward uniqueness of ultraparabolic equations.