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

TL;DR

We address the diffusive (hydrodynamic) limit of the Boltzmann equation in a half-space with a moving boundary and global rest far-field, under diffusive scaling. The authors implement a Hilbert expansion around the Maxwellian to connect kinetic dynamics to incompressible Navier–Stokes flow, identifying a Rayleigh shear profile as the formal limit through careful construction of f1 and f2 and the corresponding macroscopic variables. They establish a rigorous, finite-time construction of a Boltzmann solution about the Rayleigh profile for well-prepared initial data, controlling the remainder R. The work provides a rigorous kinetic realization of the Rayleigh problem and clarifies boundary-layer and diffusive interactions in a rarefied gas with moving boundaries, enhancing understanding of how wall motion shapes bulk flow in the diffusive regime.

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.