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

Jean-Philippe Bartier (1) , Jean Dolbeault (1) , Reinhard Illner (2) , Michal Kowalczyk (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Mathematics and Statistics
3 DIM - Depto Ingeniería Matemática

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.

Mots clés: Entropy method; relative entropy; convex entropy; contraction; large time asymptotics; stationary solutions; periodic solutions; drift-diffusion equation; logarithmic Sobolev inequality; convex Sobolev inequalities; Poincaré inequality; Caffarelli-Kohn-Nirenberg inequalities; Hardy-Poincaré inequality; Csiszár-Kulback inequality; uniqueness; large time asymptotic behavior; convergence; time-periodic solutions; singular solutions; time-dependent drift; time-dependent diffusion coefficient; entropy; entropy production method

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Philippe Bartier, Jean Dolbeault, Reinhard Illner, Michal Kowalczyk. A qualitative study of linear drift-diffusion equations with time-dependent or degenerate coefficients. Mathematical Models and Methods in Applied Sciences, 2007, 17 (3), pp.327-362. ⟨hal-00016363⟩