A new monotonicity formula for the spatially homogeneous Landau equation with Coulomb potential and its applications

Abstract

We describe a time-dependent functional involving the relative entropy and the ${\dot{H}}^1$ seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functional sheds light on the competition between dissipation and nonlinearity for this equation. It enables us to obtain new results concerning regularity/blowup issues for the Landau equation with Coulomb potential.