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Propriétés dispersives pour des équations cinétiques et applications à l’équation de Vlasov-Poisson

Delphine Salort (2008/2009)

Séminaire Équations aux dérivées partielles

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 L x , ξ 1 . 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.

Quelques résultats d’hypocoercitivité en théorie cinétique collisionnelle

Clément Mouhot (2007/2008)

Séminaire Équations aux dérivées partielles

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.

Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off.

Cédric Villani (1999)

Revista Matemática Iberoamericana

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.

Regularity in kinetic formulations via averaging lemmas

Pierre-Emmanuel Jabin, Benoît Perthame (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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 use velocity...

Regularity in kinetic formulations via averaging lemmas

Pierre-Emmanuel Jabin, Benoît Perthame (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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...

Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit

Giovanni Naldi, Lorenzo Pareschi, Giuseppe Toscani (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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...

Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit

Giovanni Naldi, Lorenzo Pareschi, Giuseppe Toscani (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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...

Stochastic approximations of the solution of a full Boltzmann equation with small initial data

Sylvie Meleard (2010)

ESAIM: Probability and Statistics

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...

The parabolic-parabolic Keller-Segel equation

Kleber Carrapatoso (2014/2015)

Séminaire Laurent Schwartz — EDP et applications

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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