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

Nadia Lekrine; Chao-Jiang Xu

doi:10.3934/krm.2009.2.647

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2009, Volume 2, Issue 4: 647-666. Doi: 10.3934/krm.2009.2.647

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

Nadia Lekrine^1, and
Chao-Jiang Xu^1,

1.
Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray

Received: March 31, 2009

Revised: July 31, 2009

Published: November 2009

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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.

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Mathematics Subject Classification: 35A05, 35B65,35D10, 42A38, 60H07, 82B40.

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