Regularity, local well-posedness, and (lack of) blow-up of solutions of 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, a full understanding of the existence, uniqueness, and qualitative behavior of solutions has remained elusive except in some simplified settings (e.g., homogeneous or perturbative). In this talk, I will report on recent progress in the application of ideas of parabolic regularity theory to this kinetic equation. Using these ideas we can, in contrast to previous results requiring boundedness of fourth derivatives of the initial data, construct solutions with low initial regularity (just $L^\infty$) 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.