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Explicit coercivity estimates for the linearized Boltzmann and Landau operators
 Comm. Partial Diff. Eq
"... Abstract. We prove explicit coercivity estimates for the linearized Boltzmann and Landau operators, for a general class of interactions including any inversepower law interactions, and hard spheres. The functional spaces of these coecivity estimates depend on the collision kernel of these operators. ..."
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Cited by 28 (3 self)
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Abstract. We prove explicit coercivity estimates for the linearized Boltzmann and Landau operators, for a general class of interactions including any inversepower law interactions, and hard spheres. The functional spaces of these coecivity estimates depend on the collision kernel of these operators. For Maxwell molecules they coincide with the spectral gap estimates. For hard potentials they are stronger and imply these spectral estimates. For soft potentials, they play the role of explicit “degenerated spectral gap ” estimates. The proofs are based on the reduction to the Maxwell case by decomposition methods. We also prove a regularity property for the linearized Boltzmann operator for non locally integrable collision kernel and for the linearized Landau operator, and we discuss the
Stability of the relativistic Maxwellian in a Collisional Plasma
 Comm. Math. Phys
, 2004
"... Abstract. The relativistic LandauMaxwell system is the most fundamental and complete model for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions and through their selfconsistent electromagnetic field. We construct the first global in ti ..."
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Cited by 24 (12 self)
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Abstract. The relativistic LandauMaxwell system is the most fundamental and complete model for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions and through their selfconsistent electromagnetic field. We construct the first global in time classical solutions. Our solutions are constructed in a periodic box and near
Relative entropies for kinetic equations in bounded domains (irreversibility, stationary solutions, uniqueness
, 2003
"... The relative entropy method describes the irreversibility of the VlasovPoisson and VlasovBoltzmannPoisson systems in bounded domains with incoming boundary conditions. Uniform in time estimates are deduced from the entropy. In some cases, these estimates are sufficient to prove the convergence of ..."
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Cited by 18 (6 self)
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The relative entropy method describes the irreversibility of the VlasovPoisson and VlasovBoltzmannPoisson systems in bounded domains with incoming boundary conditions. Uniform in time estimates are deduced from the entropy. In some cases, these estimates are sufficient to prove the convergence of the solution to a unique stationary solution, as time goes to infinity. The method is also used to analyze other types of boundary conditions such as mass and energy preserving diffuse reflection boundary conditions, and to prove the uniqueness of stationary solutions for some special collision terms. Keywords. Kinetic equations – VlasovPoisson system – Boltzmann equation – collision kernels – irreversibility – HTheorem – injection boundary conditions – diffusive boundary conditions – relative entropy – large time asymptotics – uniqueness – stationary solutions – minimization under constraints – nonlinear stability – Casimir energy – Bolza problem – plasmas – semiconductors.
Wellposedness for the spatially homogeneous LandauFermiDirac equation for hard potentials
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"... A numerical scheme for the quantum FokkerPlanckLandau equation efficient in the fluid regime ∗ ..."
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A numerical scheme for the quantum FokkerPlanckLandau equation efficient in the fluid regime ∗
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"... Equilibrium states for the LandauFermiDirac equation A kinetic collision operator of Landau type for charged FermiDirac particles is considered. Equilibrium states are rigourously determined under minimal assumptions on the distribution function of the particles. The particular structure of the c ..."
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Equilibrium states for the LandauFermiDirac equation A kinetic collision operator of Landau type for charged FermiDirac particles is considered. Equilibrium states are rigourously determined under minimal assumptions on the distribution function of the particles. The particular structure of the considered operator (strong nonlinearity and degeneracy) requires a special investigation compared to the classical Boltzmann or Landau operator. 1
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"... A numerical scheme for the quantum FokkerPlanckLandau equation efficient in the fluid regime ∗ ..."
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A numerical scheme for the quantum FokkerPlanckLandau equation efficient in the fluid regime ∗