Taylor dispersion and phase mixing in the non-cutoff Boltzmann equation on the whole space

TL;DR

This paper analyzes the long-time behavior of the non-cutoff Boltzmann equation with soft potentials, revealing phase mixing, enhanced dissipation, and Taylor dispersion effects in the weakly collisional limit on the whole space.

Contribution

It provides the first detailed analysis of phase mixing and dispersion effects for the non-cutoff Boltzmann equation near Maxwellian, including enhanced dissipation and Taylor dispersion in the weakly collisional regime.

Findings

Enhanced dissipation for high wavenumbers with decay time-scale O(1/ν^{1/(1+2s)}|k|^{2s/(1+2s)})

Taylor dispersion for low wavenumbers with decay time-scale O(ν/|k|^2)

Almost-uniform decay of macroscopic density due to Landau damping and dispersive effects

Abstract

In this paper, we describe the long-time behavior of the non-cutoff Boltzmann equation with soft potentials near a global Maxwellian background on the whole space in the weakly collisional limit (i.e. infinite Knudsen number $1/\nu\to \infty$). Specifically, we prove that for initial data sufficiently small (independent of the Knudsen number), the solution displays several dynamics caused by the phase mixing/dispersive effects of the transport operator $v \cdot \nabla_x$ and its interplay with the singular collision operator. For $x$-wavenumbers $k$ with $|k|\gg\nu$, one sees an enhanced dissipation effect wherein the characteristic decay time-scale is accelerated to $O(1/\nu^{\frac{1}{1+2s}} |k|^{\frac{2s}{1+2s}})$, where $s \in (0,1]$ is the singularity of the kernel ($s=1$ being the Landau collision operator, which is also included in our analysis); for $|k|\ll \nu$, one sees Taylor dispersion, wherein the decay is accelerated to $O(\nu/|k|^2)$. Additionally, we prove almost-uniform phase mixing estimates. For macroscopic quantities as the density $\rho$, these bounds imply almost-uniform-in-$\nu$ decay of $(t\nabla_x)^\beta \rho$ in $L^\infty_x$ due to Landau damping and dispersive decay.