From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight

TL;DR

This paper establishes exponential decay and well-posedness for the Boltzmann equation near equilibrium in weighted Sobolev spaces, connecting kinetic theory to incompressible Navier-Stokes solutions uniformly across the Knudsen number.

Contribution

It proves the generation of a $C^0$-semigroup with exponential decay for the linearized Boltzmann operator in weighted Sobolev spaces, enabling a uniform analysis of the nonlinear equation.

Findings

Exponential decay of solutions in weighted Sobolev spaces.

Uniform well-posedness of the Boltzmann equation across the Knudsen number.

Connection to incompressible Navier-Stokes solutions in these function spaces.

Abstract

We study the Boltzmann equation on the $d$-dimensional torus in a perturbative setting around a global equilibrium under the Navier-Stokes linearisation. We use a recent functional analysis breakthrough to prove that the linear part of the equation generates a $C^0$-semigroup with exponential decay in Lebesgue and Sobolev spaces with polynomial weight, independently on the Knudsen number. Finally we show a Cauchy theory and an exponential decay for the perturbed Boltzmann equation, uniformly in the Knudsen number, in Sobolev spaces with polynomial weight. The polynomial weight is almost optimal and furthermore, this result only requires derivatives in the space variable and allows to connect to solutions to the incompressible Navier-Stokes equations in these spaces.