# The Landau equation with the specular reflection boundary condition

## TL;DR

This paper proves the global stability of the Landau equation with Coulombic potential in a bounded domain with specular reflection, extending previous results and employing advanced PDE techniques under low-regularity assumptions.

## Contribution

It establishes the first global stability result for the Landau equation with boundary conditions in a general domain, using novel PDE methods and low-regularity initial data assumptions.

## Key findings

- Proves global stability of Landau equation with Coulomb potential.
- Extends well-posedness theory for kinetic Fokker-Planck equations.
- Employs advanced regularity estimates to ensure uniqueness.

## Abstract

The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box (KGH-2016). Our methods consist of the generalization of the wellposedness theory for the Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (Polidoro-Ragusa-1998) to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem.