Gevrey Smoothing for Weak Solutions of the Fully Nonlinear Homogeneous Boltzmann and Kac Equations Without Cutoff for Maxwellian Molecules

Jean-Marie Barbaroux (1, 2) , Dirk Hundertmark (3) , Tobias Ried (3) , Semjon Vugalter (3)

1 CPT - E8 Dynamique quantique et analyse spectrale
2 CPT - Centre de Physique Théorique - UMR 7332
3 KIT - Karlsruhe Institute of Technology

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.

Mots clés: Non-cutoff homogeneous Boltzmann equation; Gevrey regularity; Non-cutoff homogeneous Kac equation; Maxwellian molecules

Domaines: Physique mathématique [math-ph]; Mathématiques [math]

Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried, Semjon Vugalter. Gevrey Smoothing for Weak Solutions of the Fully Nonlinear Homogeneous Boltzmann and Kac Equations Without Cutoff for Maxwellian Molecules. Archive for Rational Mechanics and Analysis, 2017, 225 (2), pp. 601-661. ⟨10.1007/s00205-017-1101-8⟩. ⟨hal-01253130⟩