On the convergence of smooth solutions from Boltzmann to Navier-Stokes

TL;DR

This paper establishes a rigorous connection between solutions of the Boltzmann and Navier-Stokes equations, showing that the lifespan of Boltzmann solutions is at least as long as that of Navier-Stokes solutions under certain conditions.

Contribution

It provides a new proof of the convergence of Boltzmann solutions to Navier-Stokes solutions, including lifespan bounds, for general initial data in multiple dimensions.

Findings

Lifespan of Boltzmann solutions is bounded below by Navier-Stokes solutions.

Convergence holds for general initial data in 2D and 3D.

Results include well-prepared data with periodic boundary conditions.

Abstract

In this work, we are interested in the link between strong solutions of the Boltzmann and the Navier-Stokes equations. To justify this connection, our main idea is to use information on the limit system (for instance the fact that the Navier-Stokes equations are globally wellposed in two space dimensions or when the data are small). In particular we prove that the life span of the solutions to the rescaled Boltzmann equation is bounded from below by that of the Navier-Stokes system. We deal with general initial data in the whole space in dimensions 2 and 3, and also with well-prepared data in the case of periodic boundary conditions.