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A qualitative study of linear drift-diffusion equations with time-dependent or degenerate coefficients

dc.contributor.authorKowalczyk, Michal
dc.contributor.authorIllner, Reinhard
dc.contributor.authorDolbeault, Jean
HAL ID: 87
ORCID: 0000-0003-4234-2298
dc.contributor.authorBartier, Jean-Philippe
dc.date.accessioned2009-07-08T08:37:08Z
dc.date.available2009-07-08T08:37:08Z
dc.date.issued2007
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/926
dc.language.isoenen
dc.subjectlarge time asymptotic behavior
dc.subjectuniqueness
dc.subjectCsiszár-Kulback inequality
dc.subjectHardy-Poincaré inequality
dc.subjectCaffarelli-Kohn-Nirenberg inequalities
dc.subjectPoincaré inequality
dc.subjectconvex Sobolev inequalities
dc.subjectlogarithmic Sobolev inequality
dc.subjectdrift-diffusion equation
dc.subjectperiodic solutions
dc.subjectcontraction
dc.subjectconvex entropy
dc.subjectstationary solutions
dc.subjectlarge time asymptotics
dc.subjectrelative entropy
dc.subjectconvergence
dc.subjecttime-periodic solutions
dc.subjectsingular solutions
dc.subjecttime-dependent drift
dc.subjecttime-dependent diffusion coefficient
dc.subjectentropy - entropy production method
dc.subjectEntropy methoden
dc.subject.ddc519en
dc.titleA qualitative study of linear drift-diffusion equations with time-dependent or degenerate coefficientsen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherFacultad de Ciencias Fisicas y Matematicas - Universidad de Chile;Chili
dc.contributor.editoruniversityotherUniversity of Victoria;Canada
dc.description.abstractenThis 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.isversionofjnlnameMathematical Models and Methods in Applied Sciences
dc.relation.isversionofjnlvol17en
dc.relation.isversionofjnlissue3en
dc.relation.isversionofjnldate2007
dc.relation.isversionofjnlpages327-362en
dc.relation.isversionofdoihttp://dx.doi.org/10.1142/S0218202507001942
dc.identifier.urlsitehttp://hal.archives-ouvertes.fr/hal-00016363/en/en
dc.description.sponsorshipprivateouien
dc.relation.isversionofjnlpublisherWorld Scientific
dc.subject.ddclabelProbabilités et mathématiques appliquéesen


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