Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff

Abstract

Abstract For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz's space 𝒮(ℝ N ) for all (strictly positive) time. The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role. Key Words: Homogeneous Boltzmann equationNon-cutoff potentialsRegularisationSmoothness of solutions Acknowledgments We wish to thank Professor Y. Meyer for many helpful discussions in the preparation of this article. Part of the work was carried out while B.W. was visiting the CMLA, ENS Cachan, and he would like to thank L. Desvillettes and the CMLA for their hospitality. Both authors are members of the RTN network HYKE, EU contract no. HPRN-CT-2002-00282. Notes aThe referee has suggested a different proof of the results in the appendix, using a Paley-Littlewood type decomposition. We agree that this certainly gives a shorter proof. We however still present our (somewhat longer) proof, which has the advantage of relying only on completely elementary estimates.