Dr. André Schlichting (Universität Bonn): A non-local Fokker-Planck equation related to nucleation and coarsening
Mittwoch, 24.01.2018 16:15 im Raum M5
In this talk we consider a Fokker-Planck equation modeling nucleation and coarsening 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 structure involving boundary conditions.
We generalize the standard 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. Conlon, arXiv:1711.00782)