Global well-posedness for the homogeneous Landau equation

TL;DR

This paper proves global well-posedness and exponential decay to equilibrium for the homogeneous Landau equation with minimal initial regularity assumptions, using barrier arguments and parabolic regularity theory.

Contribution

It introduces new methods to establish existence and uniqueness for the Landau equation with discontinuous initial data, extending the understanding of kinetic equations.

Findings

Global existence and uniqueness for initial data with minimal regularity

Exponential decay to equilibrium established

Methods applicable to other nonlinear kinetic equations

Abstract

Global well-posedness and exponential decay to equilibrium are proved for the homogeneous Landau equation from kinetic theory. The initial distribution is only assumed to be bounded and decaying sufficiently fast at infinity. In particular, discontinuous initial configurations that might be far from equilibrium are covered. Despite the lack of a comparison principle for the equation, the proof of existence relies on barrier arguments and parabolic regularity theory. Uniqueness and decay to equilibrium are then obtained through weighted integral inequalities. Although the focus is on the spatially homogeneous case with Coulomb potential, the methods introduced here may be applied elsewhere in nonlinear kinetic theory.