On the convergence from Boltzmann to Navier-Stokes-Fourier for general initial data

TL;DR

This paper proves the convergence of solutions from the Boltzmann equation to the incompressible Navier-Stokes-Fourier system for initial data with polynomial decay, ensuring solutions do not blow up before the hydrodynamic limit.

Contribution

It introduces a novel approach combining polynomial and Gaussian decay decomposition to establish convergence for general initial data.

Findings

Proves convergence of Boltzmann solutions to Navier-Stokes-Fourier system

Demonstrates solutions do not blow up before hydrodynamic limit

Adapts decay decomposition strategy for general initial data

Abstract

In this work, we prove the convergence of strong solutions of the Boltzman equation, for initial data having polynomial decay in the velocity variable, towards those of the incompressible Navier-Stokes-Fourier system. We show in particular that the solutions of the rescaled Boltzmann equation do not blow up before their hydrodynamic limit does. This is made possible by adapting a strategy introduced by M. Briant, S. Merino and C. Mouhot of writing the solution to the Boltzmann equation as the sum a part with polynomial decay and a second one with Gaussian decay. The Gaussian part is treated with an approach reminiscent of the one used by I. Gallagher and I. Tristani.