Revista Matemática Iberoamericana


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Volume 15, Issue 2, 1999, pp. 335–352
DOI: 10.4171/RMI/259

Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off

Cédric Villani[1]

(1) UMPA, CNRS, École Normale Supérieure de Lyon, 46 allée d'Italie, 69364, LYON CEDEX 07, FRANCE

We show that in the setting of the spatially homogeneous Boltzmann equation without cut-off, the entropy dissipation associated to a function $f \in L^1(\mathbb R^N)$ yields a control of $\sqrt f$ in Sobolev norms as soon as $f$ is locally bounded below. Under this additional assumption of lower bound our result is an improvement of a recent estimate given by P.L. Lions and is optimal in a certain sense.

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Villani Cédric: Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off. Rev. Mat. Iberoamericana 15 (1999), 335-352. doi: 10.4171/RMI/259