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

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 $L_x^{\infty }$ due to Landau damping and dispersive decay.

52 pages