Gain of integrability for the Boltzmann collisional operator

Ricardo J. Alonso; Irene M. Gamba

doi:10.3934/krm.2011.4.41

Article Contents

2011, Volume 4, Issue 1: 41-51. Doi: 10.3934/krm.2011.4.41

This issue Previous Article Bounded solutions of the Boltzmann equation in the whole space Next Article A hierarchy of models related to nanoflows and surface diffusion

Gain of integrability for the Boltzmann collisional operator

Ricardo J. Alonso^1, and
Irene M. Gamba^2,

1.
Dept. of Computational & Applied Mathematics, Rice University, Houston, TX 77005-1892
2.
Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Texas 78712

Received: November 30, 2010

Revised: November 30, 2010

Published: February 2011

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Abstract

In this short note we revisit the gain of integrability property of the gain part of the Boltzmann collision operator. This property implies the $W^{l,r}_k$ regularity propagation for solutions of the associated space homogeneous initial value problem. We present a new method to prove the gain of integrability that simplifies the technicalities of previous approaches by avoiding the argument of gain of regularity estimates for the gain collisional integral. In addition our method calculates explicit constants involved in the estimates.

Keywords:

Boltzmann equation for hard potentials; Collisional Operator; Regularity in $L^p$ spaces.

Mathematics Subject Classification: 76P05, 35D05.

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