# American Institute of Mathematical Sciences

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

 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

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.
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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