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

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 (that is, infinite Knudsen number (Formula presented.)). 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 (Formula presented.) and its interplay with the singular collision operator. For (Formula presented.) -wavenumbers (Formula presented.) with (Formula presented.), one sees an enhanced dissipation effect wherein the characteristic decay time-scale is accelerated to (Formula presented.), where (Formula presented.) is the singularity of the kernel ((Formula presented.) being the Landau collision operator, which is also included in our analysis); for (Formula presented.), one sees Taylor dispersion, wherein the decay time-scale is accelerated to (Formula presented.). Additionally, we prove almost uniform phase mixing estimates. For macroscopic quantities such as the density (Formula presented.), these bounds imply almost uniform-in- (Formula presented.) decay of (Formula presented.) in (Formula presented.) due to phase mixing and dispersive decay.