The Boltzmann equation with incoming boundary condition: global solutions and Navier-Stokes limit

TL;DR

This paper proves the existence of global solutions to the Boltzmann equation with incoming boundary conditions and rigorously derives the incompressible Navier-Stokes-Fourier equations as a limit when the Knudsen number approaches zero.

Contribution

It establishes the existence of global renormalized solutions for the Boltzmann equation with boundary conditions and justifies the hydrodynamic limit to Navier-Stokes equations.

Findings

Existence of global-in-time renormalized solutions with incoming boundary conditions.

Derivation of Navier-Stokes-Fourier equations as the zero-Knudsen limit.

Convergence holds when incoming data are close to Maxwellian in boundary relative entropy.

Abstract

We consider the Boltzmann equations with cutoff collision kernels in bounded domains. For the initial data with finite physical bounds, we prove the existence of global-in-time renormalized solutions in the sense of DiPerna-Lions endowed with incoming boundary condition. Moreover, we justify the limit as the Knudsen number $\epsilon\rightarrow 0$ to Leray solutions of the incompressible Navier-Stokes-Fourier equations with homogeneous Dirichlet conditions from renormalized solutions of the scaled Boltzmann equations when the incoming data are close to the global Maxwellian in the sense of the boundary relative entropy of order $O(\epsilon^3)$