Diffusive limit of the Boltzmann equation around Rayleigh profile in the half space

TL;DR

This paper rigorously derives the diffusive limit of the Boltzmann equation in a half-space with moving boundary conditions, showing convergence to the Navier-Stokes equations with Rayleigh profile for well-prepared initial data.

Contribution

It establishes the diffusive limit of the Boltzmann equation around the Rayleigh profile in a half-space with moving boundary, using Hilbert expansion without initial singularity.

Findings

Convergence of Boltzmann solutions to Navier-Stokes with Rayleigh profile

Construction of solutions around Rayleigh profile for well-prepared data

No initial singularity over finite time intervals

Abstract

This paper concerns the diffusive limit of the time evolutionary Boltzmann equation in the half space $\mathbb{T}^2\times\mathbb{R}^+$ for a small Knudsen number $\varepsilon>0$. For boundary conditions in the normal direction, it involves diffuse reflection moving with a tangent velocity proportional to $\varepsilon$ on the wall, whereas the far field is described by a global Maxwellian with zero bulk velocity. The incompressible Navier-Stokes equations, as the corresponding formal fluid dynamic limit, admit a specific time-dependent shearing solution known as the Rayleigh profile, which accounts for the effect of the tangentially moving boundary on the flow at rest in the far field. Using the Hilbert expansion method, for well-prepared initial data we construct the Boltzmann solution around the Rayleigh profile without initial singularity over any finite time interval.