Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation

Pages: 647 - 666, Volume 2, Issue 4, December 2009

doi:10.3934/krm.2009.2.647       Abstract        Full Text (246.5K)       Related Articles

Nadia Lekrine - Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray, France (email)
Chao-Jiang Xu - Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray, France (email)

Abstract: In this work, we consider a spatially homogeneous Kac's equation with a non cutoff cross section. We prove that the weak solution of the Cauchy problem is in the Gevrey class for positive time. This is a Gevrey regularizing effect for non smooth initial datum. The proof relies on the Fourier analysis of Kac's operators and on an exponential type mollifier.

Keywords:  Non-cutoff Kac's equation, Boltzmann equation, Gevrey regularizing effect, Cauchy problem, Fourier analysis.
Mathematics Subject Classification:  35A05, 35B65,35D10, 42A38, 60H07, 82B40.

Received: April 2009;      Revised: August 2009;      Published: October 2009.