We show that in the setting of the spatially homogeneous Boltzmann equation without cut-off, the entropy dissipation associated to a function yields a control of in Sobolev norms as soon as 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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Cédric Villani, Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off. Rev. Mat. Iberoam. 15 (1999), no. 2, pp. 335–352DOI 10.4171/RMI/259