Special Session 47: Singular limits in elliptic and parabolic PDEs

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

Michele Dolce
Co-Author(s):    Jacob Bedrossian, Michele Coti Zelati
Consider the non-cutoff Boltzmann equation with soft potentials on the whole space in the large Knudsen number $\operatorname{Kn}\gg 1$ regime, describing for instance molecules in the upper atmosphere. We study quantitative stability properties of a global Maxwellian background. Specifically, we prove that for initial data sufficiently small (independent of $\operatorname{Kn}$), 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. One consequence is the \textit{Taylor dispersion}, showing that the perturbation decay on a time-scale $O(1)$. This is a faster relaxation time-scale compared to the $O(\operatorname{Kn})$ expected when only the collision operator is present. Additionally, we prove almost-uniform phase mixing estimates. For macroscopic quantities as the density $\rho$, these bounds imply almost-uniform-in-$v$ decay of $(t\nabla_x)\rho $ in $L^\infty_x$ due to Landau damping and dispersive decay.