Gevrey Smoothing for Weak Solutions of the Fully Nonlinear Homogeneous Boltzmann and Kac Equations Without Cutoff for Maxwellian Molecules
Résumé
It has long been suspected that the non-cutoff Boltzmann operator has similar coerciv-ity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular , the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in L 1 2 (R d) ∩ L log L(R d), i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.
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