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Niveau: Supérieur, Doctorat, Bac+8

ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR MAXWELLIAN MOLECULES C. VILLANI Abstract. We establish a simplified form for the Landau equa- tion with Maxwellian-type molecules. We study in detail the Cauchy problem associated to this equation, and some qualitative features of the solution. Explicit solutions are also given. Contents 1. Introduction 1 2. Simplified Expression 3 3. The isotropic case : Fokker-Planck equation 5 4. The collision operator and the Cauchy problem 8 5. Final decomposition of the Landau collision operator 14 6. Weak formulations and applications : time-evolution of the moments, Maxwellian tails, energy estimates 15 7. Positivity 21 8. Long-time behaviour 22 9. Self-similar solutions 26 Appendix A. Inverse Fourier Transform of e?|?|2/2|?|? 27 Appendix B. The Nikolskii Transform 28 References 29 1. Introduction The Landau equation (also called sometimes Fokker-Planck) is a common kinetic model in plasma physics. It is a nonlinear partial differential equation where the unknown function, f , is the density of a “gas” in the phase space of all positions and velocities of “particles”. We shall assume that these vary in RN , N ≥ 2. In the case of a gas composed of a single species and if we assume that the density function is spatially homogeneous, ie does not depend on the position but only 1

ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR MAXWELLIAN MOLECULES C. VILLANI Abstract. We establish a simplified form for the Landau equa- tion with Maxwellian-type molecules. We study in detail the Cauchy problem associated to this equation, and some qualitative features of the solution. Explicit solutions are also given. Contents 1. Introduction 1 2. Simplified Expression 3 3. The isotropic case : Fokker-Planck equation 5 4. The collision operator and the Cauchy problem 8 5. Final decomposition of the Landau collision operator 14 6. Weak formulations and applications : time-evolution of the moments, Maxwellian tails, energy estimates 15 7. Positivity 21 8. Long-time behaviour 22 9. Self-similar solutions 26 Appendix A. Inverse Fourier Transform of e?|?|2/2|?|? 27 Appendix B. The Nikolskii Transform 28 References 29 1. Introduction The Landau equation (also called sometimes Fokker-Planck) is a common kinetic model in plasma physics. It is a nonlinear partial differential equation where the unknown function, f , is the density of a “gas” in the phase space of all positions and velocities of “particles”. We shall assume that these vary in RN , N ≥ 2. In the case of a gas composed of a single species and if we assume that the density function is spatially homogeneous, ie does not depend on the position but only 1

- landau equation
- ?ij ?
- bi ?
- landau collision
- spatially homogeneous
- ∆f ? ∑
- landau equa- tion only

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Published by | mijec |

Reads | 74 |

Language | English |

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