# The Boltzmann equation with an external force on the torus: Incompressible Navier-Stokes-Fourier hydrodynamical limit

## TL;DR

This paper establishes a uniform-in-Knudsen-number theory for the Boltzmann equation with external forces on the torus, leading to the derivation of the incompressible Navier-Stokes-Fourier system without smallness or decay assumptions on the force.

## Contribution

It develops a robust, uniform-in-Knudsen-number theory for the Boltzmann equation with general external forces, enabling derivation of hydrodynamic limits without smallness or decay conditions.

## Key findings

- Established local-in-time Cauchy theories independent of the Knudsen number.
- Proved existence around a time-dependent Maxwellian capturing force-induced fluctuations.
- Derived the incompressible Navier-Stokes-Fourier system with external force from Boltzmann equation.

## Abstract

We study the Boltzmann equation with external forces, not necessarily deriving from a potential, in the incompressible Navier-Stokes perturbative regime. On the torus, we establish local-in-time, for any time, Cauchy theories that are independent of the Knudsen number in Sobolev spaces. The existence is proved around a time-dependent Maxwellian that behaves like the global equilibrium both as time grows and as the Knudsen number decreases. We combine hypocoercive properties of linearized Boltzmann operators with linearization around a time-dependent Maxwellian that catches the fluctuations of the characteristics trajectories due to the presence of the force. This uniform theory is sufficiently robust to derive the incompressible Navier-Stokes-Fourier system with an external force from the Boltzmann equation. Neither smallness, nor time-decaying assumption is required for the external force, nor a gradient form, and we deal with general hard potential and cut-off Boltzmann kernels. As a by-product the latest general theories for unit Knudsen number when the force is sufficiently small and decays in time are recovered.