Gevrey smoothing of weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules

We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with finite mass, energy and entropy, that is, $f_0 \in L^1_2(\R^d) \cap L \log L(\R^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity.

This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers.

(Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)