Uncertainty principle and kinetic equations |
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Authors: | R Alexandre Y Morimoto |
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Institution: | a IRENAv, Research Institute, French Naval Academy, Brest-Lanvéoc, France b Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan c Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong, PR China d Université de Rouen, UMR 6085-CNRS, Mathématiques Avenue de l'Université, BP 12, 76801 Saint Etienne du Rouvray, France e Department of Mathematics, City University of Hong Kong, Hong Kong, PR China |
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Abstract: | A large number of mathematical studies on the Boltzmann equation are based on the Grad's angular cutoff assumption. However, for particle interaction with inverse power law potentials, the associated cross-sections have a non-integrable singularity corresponding to the grazing collisions. Smoothing properties of solutions are then expected. On the other hand, the uncertainty principle, established by Heisenberg in 1927, has been developed so far in various situations, and it has been applied to the study of the existence and smoothness of solutions to partial differential equations. This paper is the first one to apply this celebrated principle to the study of the singularity in the cross-sections for kinetic equations. Precisely, we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations. In particular, we give some estimates on the regularity of solutions in Sobolev spaces w.r.t. all variables for both linearized and nonlinear space inhomogeneous Boltzmann equations without angular cutoff, as well as the linearized space inhomogeneous Landau equation. |
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Keywords: | Uncertainty principle Kinetic equations Microlocal analysis Non-cutoff cross-sections Boltzmann equations Landau equation |
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