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

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June 2010, 3(2): 335-351. doi: 10.3934/krm.2010.3.335

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, United States
2.	Department of Mathematics, University of Maryland, College Park, MD 20742, United States

Received September 2009 Revised November 2009 Published May 2010

Abstract
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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.

Keywords: Compactness, Fredholm property, linearized Boltzmann operator..

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: C. David Levermore, Weiran Sun. Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels. Kinetic & Related Models, 2010, 3 (2) : 335-351. doi: 10.3934/krm.2010.3.335

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