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In this paper, we describe the long-time behavior of the non-cutoff Boltzmannequation with soft potentials near a global Maxwellian background on the wholespace 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 dynamicscaused 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 effectwherein 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 thesingularity of the kernel ($s=1$ being the Landau collision operator, which isalso included in our analysis); for $|k|\ll \nu$, one sees Taylor dispersion,wherein the decay is accelerated to $O(\nu/|k|^2)$. Additionally, we provealmost-uniform phase mixing estimates. For macroscopic quantities as thedensity $\rho$, these bounds imply almost-uniform-in-$\nu$ decay of$(t\nabla_x)^\beta \rho$ in $L^\infty_x$ due to Landau damping and dispersivedecay.

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