dc.contributor.author Kowalczyk, Michal dc.contributor.author Illner, Reinhard dc.contributor.author Dolbeault, Jean dc.contributor.author Bartier, Jean-Philippe dc.date.accessioned 2009-07-08T08:37:08Z dc.date.available 2009-07-08T08:37:08Z dc.date.issued 2007 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/926 dc.language.iso en en dc.subject large time asymptotic behavior dc.subject uniqueness dc.subject Csiszár-Kulback inequality dc.subject Hardy-Poincaré inequality dc.subject Caffarelli-Kohn-Nirenberg inequalities dc.subject Poincaré inequality dc.subject convex Sobolev inequalities dc.subject logarithmic Sobolev inequality dc.subject drift-diffusion equation dc.subject periodic solutions dc.subject contraction dc.subject convex entropy dc.subject stationary solutions dc.subject large time asymptotics dc.subject relative entropy dc.subject convergence dc.subject time-periodic solutions dc.subject singular solutions dc.subject time-dependent drift dc.subject time-dependent diffusion coefficient dc.subject entropy - entropy production method dc.subject Entropy method en dc.subject.ddc 519 en dc.title A qualitative study of linear drift-diffusion equations with time-dependent or degenerate coefficients en dc.type Article accepté pour publication ou publié dc.contributor.editoruniversityother Facultad de Ciencias Fisicas y Matematicas - Universidad de Chile;Chili dc.contributor.editoruniversityother University of Victoria;Canada dc.description.abstracten This paper is concerned with entropy methods for linear drift-diffusion equations with explicitly time-dependent or degenerate coefficients. Our goal is to establish a list of various qualitative properties of the solutions. The motivation for this study comes from a model for molecular motors, the so-called Brownian ratchet, and from a nonlinear equation arising in traffic flow models, for which complex long time dynamics occurs. General results are out of the scope of this paper, but we deal with several examples corresponding to most of the expected behaviors of the solutions. We first prove a contraction property for general entropies which is a useful tool for uniqueness and for the convergence to some eventually time-dependent large time asymptotic solutions. Then we focus on power law and logarithmic relative entropies. When the diffusion term is of the type $\nabla(|x|^\alpha\,\nabla\cdot)$, we prove that the inequality relating the entropy with the entropy production term is a Hardy-Poincaré type inequality, that we establish. Here we assume that $\alpha\in (0,2]$ and the limit case $\alpha=2$ appears as a threshold for the method. As a consequence, we obtain an exponential decay of the relative entropies. In the case of time-periodic coefficients, we prove the existence of a unique time-periodic solution which attracts all other solutions. The case of a degenerate diffusion coefficient taking the form $|x|^\alpha$ with $\alpha>2$ is also studied. The Gibbs state exhibits a non integrable singularity. In this case concentration phenomena may occur, but we conjecture that an additional time-dependence restores the smoothness of the asymptotic solution. en dc.relation.isversionofjnlname Mathematical Models and Methods in Applied Sciences dc.relation.isversionofjnlvol 17 en dc.relation.isversionofjnlissue 3 en dc.relation.isversionofjnldate 2007 dc.relation.isversionofjnlpages 327-362 en dc.relation.isversionofdoi http://dx.doi.org/10.1142/S0218202507001942 dc.identifier.urlsite http://hal.archives-ouvertes.fr/hal-00016363/en/ en dc.description.sponsorshipprivate oui en dc.relation.isversionofjnlpublisher World Scientific dc.subject.ddclabel Probabilités et mathématiques appliquées en
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