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Non linear Diffusions as Limit of Kinetic Equations with Relaxation Collision Kernels

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Abstract

Kinetic transport equations with a given confining potential and non-linear relaxation type collision operators are considered. General (monotone) energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation. Existence and uniqueness of solutions is proved for initial data bounded by equilibrium distributions. The diffusive macroscopic limit is carried out using compensated compactness theory. The results are drift-diffusion equations with non linear diffusion. The most notable examples are of the form \(\partial_t \rho=\nabla\cdot (\nabla \rho^m + \rho\,\nabla V)\) , ranging from porous medium equations to fast diffusion, with the exponent satisfying \(0 < m < 5/3\) in \({\mathbb{R}}^3\) .

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Authors and Affiliations

  1. Ceremade, Université Paris Dauphine, Place de Lattre de Tassigny, F-75775, Paris Cédex 16, France

    Jean Dolbeault

  2. Faculty of Mathematics, University of Vienna, Nordbergstr. 15, A-1090, Wien, Austria

    Peter Markowich, Dietmar Oelz & Christian Schmeiser

  3. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, 4040, Linz, Austria

    Christian Schmeiser

Authors
  1. Jean Dolbeault
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  2. Peter Markowich
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  3. Dietmar Oelz
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  4. Christian Schmeiser
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Correspondence to Jean Dolbeault.

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Communicated by Y. Brenier.

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Dolbeault, J., Markowich, P., Oelz, D. et al. Non linear Diffusions as Limit of Kinetic Equations with Relaxation Collision Kernels. Arch Rational Mech Anal 186, 133–158 (2007). https://doi.org/10.1007/s00205-007-0049-5

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  • DOI: https://doi.org/10.1007/s00205-007-0049-5

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