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On considère l’équation de Vlasov-Poisson en dimension 3. On montre des résultats d’existence et d’unicité de solutions faibles de l’équation de Vlasov-Poisson avec densité bornée pour des données initiales ayant strictement moins de six moments dans . La preuve est basée sur une nouvelle approche qui consiste à établir des effets de moments a priori pour des équations de transport avec des termes de force peu réguliers.
Nous présentons une introduction à un nouveau champ de recherche, l’hypocoercitivité. Nous énonçons quelques résultats obtenus récemment avec différents co-auteurs (Lukas Neumann, Jean Dolbeault, Christian Schmeiser) dans le cas des équations cinétiques collisionnelles, en particulier pour les équations de type Boltzmann. Puis nous présentons quelques perspectives de recherche à plus long terme, dans le but de dégager une théorie unifiée de l’hypocoercitivité en théorie cinétique collisionnelle.
We show that in the setting of the spatially homogeneous Boltzmann equation without cut-off, the entropy dissipation associated to a function f ∈ L1(RN) yields a control of √f in Sobolev norms as soon as f 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.
We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity...
We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to...
In this paper we introduce numerical schemes for a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. In particular, we study the numerical passage of the Boltzmann equation with singular kernel to nonlinear friction equations in the so-called quasi elastic limit. To this aim we introduce a Fourier spectral method for the Boltzmann equation [25, 26] and show that the kernel modes that define the spectral method have the correct...
In this paper we introduce numerical schemes for a
one-dimensional kinetic model of the Boltzmann equation with
dissipative collisions and variable coefficient of restitution. In
particular, we study the numerical passage of the Boltzmann
equation with singular kernel to nonlinear friction equations in
the so-called quasi elastic limit. To this aim we introduce a
Fourier spectral method for the Boltzmann equation [CITE]
and show that the kernel modes that define the spectral method
have the correct...
This paper gives an approximation of the solution of the Boltzmann
equation by stochastic interacting particle systems in a case of
cut-off collision operator and small initial data. In this case,
following the ideas of Mischler and Perthame, we prove the existence
and uniqueness of the solution of this equation and also the existence
and uniqueness of the solution of the associated nonlinear martingale
problem.
Then, we first delocalize the interaction by considering a mollified
Boltzmann...
I present in this note recent results on the uniqueness and stability for the parabolic-parabolic Keller-Segel equation on the plane, obtained in collaboration with S. Mischler in [11].
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