Blow-up criteria and smoothing for the Landau equation
The Landau equation is a mesoscopic model in plasma physics that describes the evolution in phase-space of the density of colliding particles. Due to the non-local, non-linear terms in the equation, an understanding of the existence, uniqueness, and qualitative behavior of solutions has remained elusive except in homogeneous (i.e. x-independent) or perturbative settings. In this talk, I will report on recent process to apply ideas of parabolic regularity theory to this kinetic setting. In particular, we construct solutions with low initial regularity and show they are smooth and bounded for all time as long as the mass and energy densities remain bounded. This is a joint work with S. Snelson and A. Tarfulea.