Compactness of the gain parts of the linearized       Boltzmann operator with weakly cutoff kernels

C. David Levermore; Weiran Sun

doi:10.3934/krm.2010.3.335

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2010, Volume 3, Issue 2: 335-351. Doi: 10.3934/krm.2010.3.335

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Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels

C. David Levermore^1, and
Weiran Sun^2,

1.
Department of Mathematics & Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742
2.
Department of Mathematics, University of Maryland, College Park, MD 20742

Received: August 31, 2009

Revised: October 31, 2009

Published: May 2010

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Abstract

We prove an $L^p$ compactness result for the gain parts of the linearized Boltzmann collision operator associated with weakly cutoff collision kernels that derive from a power-law intermolecular potential. We replace the Grad cutoff assumption previously made by Caflisch [1], Golse and Poupaud [7], and Guo [11] with a weaker local integrability assumption. This class includes all classical kernels to which the DiPerna-Lions theory applies that derive from a repulsive inverse-power intermolecular potential. In particular, our approach allows the treatment of both hard and soft potential cases.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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