Well-posedness of the Lenard-Balescu equation with smooth interactions

TL;DR

This paper proves the well-posedness of the Lenard-Balescu equation with smooth interactions, extending the mathematical understanding of this fundamental kinetic equation beyond Coulomb interactions.

Contribution

It establishes global and local well-posedness results for the Lenard-Balescu equation with smooth potentials, including convergence to equilibrium and validation of the Landau approximation.

Findings

Global well-posedness near equilibrium

Local well-posedness away from equilibrium

Convergence to equilibrium and Landau approximation validity

Abstract

The Lenard-Balescu equation was formally derived in the 1960s as the fundamental description of the collisional process in a spatially homogeneous system of interacting particles. It can be viewed as correcting the standard Landau equation by taking into account collective screening effects. Due to the reputed complexity of the Lenard-Balescu equation in case of Coulomb interactions, its mathematical theory has remained void apart from the linearized setting. In this contribution, we focus on the case of smooth interactions and we show that dynamical screening effects can then be handled perturbatively. Taking inspiration from the Landau theory, we establish global well-posedness close to equilibrium, local well-posedness away from equilibrium, and we discuss the convergence to equilibrium and the validity of the Landau approximation.