Fokker-Planck equation in bounded domain

Chupin, Laurent

Annales de l'Institut Fourier, Tome 60 (2010) no. 1, pp. 217-255.

On étudie l’existence et l’unicité de solution $\varphi$ à l’équation de Fokker-Planck linéaire $-\Delta \varphi +div\left(\varphi \mathbf{F}\right)=f$ sur un domaine borné de ${ℝ}^d$ lorsque $\mathbf{F}$ est un champ de vecteurs “confinant” comme par exemple l’inverse de la distance au bord. Une illustration des résultats obtenus est donnée dans le cadre de la mécanique des fluides et des écoulements de polymères.

We study the existence and the uniqueness of a solution $\varphi$ to the linear Fokker-Planck equation $-\Delta \varphi +div\left(\varphi \mathbf{F}\right)=f$ in a bounded domain of ${ℝ}^d$ when $\mathbf{F}$ is a “confinement” vector field. This field acting for instance like the inverse of the distance to the boundary. An illustration of the obtained results is given within the framework of fluid mechanics and polymer flows.

Classification : 35J25, 35Q35, 35R60, 76A05, 82D60
Keywords: Fokker-Planck equation, Bounded domain, Stationary solution, Confinement, Fluid mechanics, Polymer flows
Mot clés : équation de Fokker-Planck, domaine borné, confinement, mécanique des fluides, écoulement de polymères

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