Some A priori estimates for the homogeneous Landau equation with soft potentials

Abstract

This paper deals with the derivation of some á priori estimates for the homogeneous Landau equation with soft potentials. Using the coercivity of the Landau operator for soft potentials, we prove a global estimate of weak solutions in $L^2$ space without any smallness assumption on the initial data for $-2<\gamma <0$. For the stronger case $-3\le \gamma \le -2$, which covers in particular the Coulomb case, we get such a global estimate, but in some weighted $L^2$ space and under a smallness assumption on initial data.