A non-local Fokker-Planck equation related to the Becker-Döring model

In this talk we consider a Fokker-Planck equation modelling nucleation of clusters very similar to the classical Becker-Döring equation. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system. In this way the equation has formally the structure of a McKean-Vlasov equation, but with a non-local boundary condition.

We briefly discuss the well-posedness and regularity properties of the Cauchy-Problem. Here the main difficulty is to improve basic a priori regularity properties of the non-local order parameter. The main part of the talk focuses on the longtime behaviour of the system. The system posses a free energy, which strictly decreases along the evolution and leads to a gradient flow structure involving boundary conditions.

We generalize the entropy method based on suitable weighted logarithmic Sobolev inequalities and interpolation estimates. In this way, we obtain an explicit characterization of the convergence to equilibrium with algebraic or even exponential rates depending on the particular assumptions on the vector fields, diffusivity and initial data.

(joint work with J. Canizo, J. Conlon)