Application de la décomposition de Littlewood-Paley à la régularité pour des équations cinétiques de type Boltzmann

Abstract : We study the regularity of kinetic equations of Boltzmann type. We use essentielly Littlewood-Paley method from harmonic analysis, consisting mainly in working with dyadic annulus. We shall mainly concern with the homogeneous case, where the solution f(t,x,v) depends only on the time t and on the velocities v, while working with realistic and singular cross-sections (non cutoff).
In the first part, we study the particular case of Maxwellian molecules. Under this hypothesis, the structure of the Boltzmann operator and his Fourier transform write in a simple form. We show a global C^\infty regularity.
Then, we deal with the case of general cross-sections with "hard potential". We are interested in the Landau equation which is limit equation to the Boltzmann equation, taking in account grazing collisions. We prove that any weak solution belongs to Schwartz space S. We demonstrate also a similar regularity for the case of Boltzmann equation. Let us note that our method applies directly for all dimensions, and proofs are often simpler compared to other previous ones.
Finally, we finish with Boltzmann-Dirac equation. In particular, we adapt the result of regularity obtained in Alexandre, Desvillettes, Wennberg and Villani work, using the dissipation rate connected with Boltzmann-Dirac equation.
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https://tel.archives-ouvertes.fr/tel-00195091
Contributor : Mouhamad El Safadi <>
Submitted on : Sunday, December 9, 2007 - 2:27:45 PM
Last modification on : Thursday, May 3, 2018 - 3:32:06 PM
Long-term archiving on: Monday, April 12, 2010 - 6:42:12 AM

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  • HAL Id : tel-00195091, version 1

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Mouhamad El Safadi. Application de la décomposition de Littlewood-Paley à la régularité pour des équations cinétiques de type Boltzmann. Mathématiques [math]. Université d'Orléans, 2007. Français. ⟨tel-00195091⟩

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