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Relative Entropies for Kinetic Equations in Bounded Domains (Irreversibility, Stationary Solutions, Uniqueness)

Ben Abdallah, Naoufel; Dolbeault, Jean (2003), Relative Entropies for Kinetic Equations in Bounded Domains (Irreversibility, Stationary Solutions, Uniqueness), Archive for Rational Mechanics and Analysis, 168, 4, p. 253-298. http://dx.doi.org/10.1007/s00205-002-0239-0

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Type
Article accepté pour publication ou publié
Date
2003
Journal name
Archive for Rational Mechanics and Analysis
Volume
168
Number
4
Publisher
Springer
Pages
253-298
Publication identifier
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Author(s)
Ben Abdallah, Naoufel
Dolbeault, Jean cc
Abstract (EN)
The relative entropy method describes the irreversibility of the Vlasov-Poisson and VlasovBoltzmann -Poisson 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.
Subjects / Keywords
Kinetic equations; Vlasov-Poisson system; Boltzmann equation; collision kernels; irreversibility; H-Theorem; injection boundary conditions; diffusive boundary conditions; relative entropy; large time asymptotics; uniqueness; stationary solutions; minimization under constraints; nonlinear stability; Casimir energy; Bolza problem; plasmas; semi-conductors