Relative entropy and slightly compressible Navier-Stokes dynamics of the Boltzmann equation

TL;DR

This paper demonstrates, at a formal level, how solutions of the Boltzmann equation converge to those of the slightly compressible Navier-Stokes system using the relative entropy method, elucidating the convergence rate in a three-dimensional periodic setting.

Contribution

It extends the relative entropy method to analyze the convergence of Boltzmann solutions to the compressible Navier-Stokes equations with small Mach number.

Findings

Formal convergence of Boltzmann to compressible Navier-Stokes solutions.

Characterization of the entropy evolution relative to local Maxwellian.

Quantitative convergence rate from Boltzmann to Navier-Stokes.

Abstract

This paper shows that, in the formal level, the convergence of solutions of Boltzmann equation to solutions of the compressible Navier-Stokes system with small Mach number over the three-dimensional periodic domain $\mathbb{T}^3$, using the relative entropy method originated from Bardos, Golse, Levermore [{\em Comm. Pure Appl. Math.} {\bf 46} (1993) 667--753] and Yau [{\em Lett. Math. Phys.} {\bf 22} (1991) 63--80]. We discuss the evolution of the entropy which is relative to the local Maxwellian governed by the solution of slightly compressible Navier-Stokes system. This characterizes the convergence rate from Boltzmann equation to the incompressible Navier-Stokes system.