American Institute of Mathematical Sciences

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September  2010, 3(3): 445-456. doi: 10.3934/krm.2010.3.445

The uniformly heated inelastic Boltzmann equation in Fourier space

Ralf Kirsch 1, and  Sergej Rjasanow 2,

1. 

Department of Flow and Material Simulation, Fraunhofer ITWM, Fraunhofer-Platz 1, D-67663 Kaiserslautern, Germany
2. 

Department of Mathematics, Saarland University, P.O. Box 15 11 50, D-66041 Saarbrücken, Germany

Received  January 2010 Revised  May 2010 Published  July 2010

In this article, we present an alternative formulation of the Boltzmann equation for diffusively driven granular media. The equation is considered with minimal a priori assumptions, i.e. in weak form in the sense of tempered distributions. Using shifted test functions and the Fourier transform, it is seen that the transformed problem contains only a threefold integral. For constant restitution coefficients and the variable hard spheres model, explicit expressions of the integral kernel in the transformed collision operator are obtained. The version of the equation derived here is a true extension of the elastic case. Some well-known results for Maxwell molecules with inelastic interactions are recovered.
Keywords: Granular materials, homogeneous Boltzmann equation..
Mathematics Subject Classification: 82C40, 76P0.
Citation: Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445
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