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Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift

David J. Knezevic, Endre Süli (2009)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to + ∞ along the boundary ∂ D of the computational domain D . Using a symmetrization...

Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift

David J. Knezevic, Endre Süli (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to +∞ along the boundary ∂D of the computational domain D. Using a symmetrization...

Spreading and vanishing in nonlinear diffusion problems with free boundaries

Yihong Du, Bendong Lou (2015)

Journal of the European Mathematical Society

We study nonlinear diffusion problems of the form u t = u x x + f ( u ) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special f ( u ) of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any f ( u ) which is C 1 and satisfies f ( 0 ) = 0 , we show that the omega limit set ω ( u ) of every bounded positive solution is determined by a stationary solution....

Stability and instability of equilibria on singular domains

Maria Gokieli, Nicolas Varchon (2009)

Banach Center Publications

We show existence of nonconstant stable equilibria for the Neumann reaction-diffusion problem on domains with fractures inside. We also show that the stability properties of all hyperbolic equilibria remain unchanged under domain perturbation in a quite general sense, covered by the theory of Mosco convergence.

Stability in nonlinear evolution problems by means of fixed point theorems

Jaromír J. Koliha, Ivan Straškraba (1997)

Commentationes Mathematicae Universitatis Carolinae

The stabilization of solutions to an abstract differential equation is investigated. The initial value problem is considered in the form of an integral equation. The equation is solved by means of the Banach contraction mapping theorem or the Schauder fixed point theorem in the space of functions decreasing to zero at an appropriate rate. Stable manifolds for singular perturbation problems are compared with each other. A possible application is illustrated on an initial-boundary-value problem for...

Stochastic calculus and degenerate boundary value problems

Patrick Cattiaux (1992)

Annales de l'institut Fourier

Consider the boundary value problem (L.P): ( h - A ) u = f in D , ( v - Γ ) u = g on ∂ D where A is written as A = 1 / 2 ∑ i = 1 m Y i 2 + Y 0 , and Γ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields Y i ( 0 ≤ i ≤ m ), for regular open sets D with a non-characteristic boundary.Our study lies on the stochastic representation of u and uses the stochastic calculus of variations for the ( A , Γ ) -diffusion process...

Sur des problèmes d’asservissements stratigraphiques

Gérard Gagneux, Guy Vallet (2002)

ESAIM: Control, Optimisation and Calculus of Variations

On expose les difficultés d’ordre mathématique que posent des modèles récents de sédimentation-érosion de bassins élaborés par l’Institut Français du Pétrole et fondés sur la prise en compte de diverses contraintes d’unilatéralité. On présente quelques résultats partiels théoriques et des directions de recherche pour la résolution d’un problème inverse posé par l’étude stratigraphique d’une colonne monolithologique.

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