# Trend to the equilibrium for the Fokker-Planck system with a strong external magnetic field

## TL;DR

This paper studies the Fokker-Planck equation under a strong magnetic field, proving global existence and exponential convergence to equilibrium in various functional spaces.

## Contribution

It extends previous methods to establish convergence to equilibrium for the Fokker-Planck system with a magnetic field in broader spaces.

## Key findings

- Global solutions near Maxwellian are constructed.
- Exponential convergence to equilibrium is proven.
- Results are extended to Lebesgue and Sobolev spaces.

## Abstract

We consider the Fokker-Planck equation with a strong external magnetic field. Global-in-time solutions are built near the Maxwellian, the global equilibrium state for the system. Moreover, we prove the convergence to equilibrium at exponential rate. The results are first obtained on spaces with an exponential weight. Then they are extended to larger functional spaces, like the Lebesgue space and the Sobolev space with polynomial weight, by the method of factorization and enlargement of the functional space developed in [Gualdani, Mischler, Mouhot, 2017].