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

TL;DR

This paper establishes a priori estimates for the homogeneous Landau equation with soft potentials, demonstrating global bounds for weak solutions in specific function spaces without initial data smallness assumptions.

Contribution

It provides new global a priori estimates for weak solutions of the Landau equation with soft potentials, including the Coulomb case, under different conditions.

Findings

Global $L^2$ estimates for $-2<\, ext{gamma}<0$ without smallness assumptions.

Global estimates in weighted $L^2$ spaces for $-3\, ext{gamma}\, ext{leq}-2$ with small initial data.

Extension of coercivity techniques to derive these estimates.

Abstract

This paper deals with the derivation of some \'a 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 \leq \gamma \leq -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.