A qualitative study of linear driftdiffusion equations with timedependent or degenerate coefficients
Kowalczyk, Michal; Illner, Reinhard; Dolbeault, Jean; Bartier, JeanPhilippe (2007), A qualitative study of linear driftdiffusion equations with timedependent or degenerate coefficients, Mathematical Models and Methods in Applied Sciences, 17, 3, p. 327362. http://dx.doi.org/10.1142/S0218202507001942
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Article accepté pour publication ou publiéExternal document link
http://hal.archivesouvertes.fr/hal00016363/en/Date
2007Journal name
Mathematical Models and Methods in Applied SciencesVolume
17Number
3Publisher
World Scientific
Pages
327362
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Show full item recordAbstract (EN)
This paper is concerned with entropy methods for linear driftdiffusion equations with explicitly timedependent 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 socalled 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 timedependent 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 HardyPoincaré 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 timeperiodic coefficients, we prove the existence of a unique timeperiodic 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 timedependence restores the smoothness of the asymptotic solution.Subjects / Keywords
large time asymptotic behavior; uniqueness; CsiszárKulback inequality; HardyPoincaré inequality; CaffarelliKohnNirenberg inequalities; Poincaré inequality; convex Sobolev inequalities; logarithmic Sobolev inequality; driftdiffusion equation; periodic solutions; contraction; convex entropy; stationary solutions; large time asymptotics; relative entropy; convergence; timeperiodic solutions; singular solutions; timedependent drift; timedependent diffusion coefficient; entropy  entropy production method; Entropy methodRelated items
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