Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules

Alexander V. Bobylev; Irene M. Gamba

doi:10.3934/krm.2017023

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2017, Volume 10, Issue 3: 573-585. Doi: 10.3934/krm.2017023

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Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules

Alexander V. Bobylev^1, and
Irene M. Gamba^2,

1.
Keldysh Institute for Applied Mathematics RAS, Miusskaya Sq. 4, 125047 Moscow, RF, Russia
2.
Department of Mathematics and ICES, The University of Texas at Austin, 1 University Station C0200, Texas 78712, USA

Received: July 2016

Revised: August 2016

Published: September 2017

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Abstract

We consider solutions to the initial value problem for the spatially homogeneous Boltzmann equation for pseudo-Maxwell molecules and show uniform in time propagation of upper Maxwellians bounds if the initial distribution function is bounded by a given Maxwellian. First we prove the corresponding integral estimate and then transform it to the desired local estimate. We remark that propagation of such upper Maxwellian bounds were obtained by Gamba, Panferov and Villani for the case of hard spheres and hard potentials with angular cut-off. That manuscript introduced the main ideas and tools needed to prove such local estimates on the basis of similar integral estimates. The case of pseudo-Maxwell molecules needs, however, a special consideration performed in the present paper.

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Mathematics Subject Classification: 76P05, 35D05.

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References

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