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Stability and Uniqueness for the Spatially Homogeneous Boltzmann Equation with Long-Range Interactions

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Abstract

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering all physical collision kernels). These estimates are conditional on some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non-cutoff and non-mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions.

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Authors and Affiliations

  1. CMLA, ÉNS Cachan, 61, avenue du Président Wilson, 94235, Cachan Cedex, France

    Laurent Desvillettes

  2. CNRS & CEREMADE, Univ. Paris IX-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Clément Mouhot

Authors
  1. Laurent Desvillettes
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  2. Clément Mouhot
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Correspondence to Clément Mouhot.

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Communicated by P.L. Lions

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Desvillettes, L., Mouhot, C. Stability and Uniqueness for the Spatially Homogeneous Boltzmann Equation with Long-Range Interactions. Arch Rational Mech Anal 193, 227–253 (2009). https://doi.org/10.1007/s00205-009-0233-x

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  • DOI: https://doi.org/10.1007/s00205-009-0233-x

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