Nonlinear inhomogeneous Fokker–Planck models: Energetic-variational structures and long-time behavior

Abstract

Inspired by the modeling of grain growth in polycrystalline materials, we consider a nonlinear Fokker–Plank model, with inhomogeneous diffusion and with variable mobility parameters. We develop large time asymptotic analysis of such nonstandard models by reformulating and extending the classical entropy method, under the assumption of periodic boundary condition. In addition, illustrative numerical tests are presented to highlight the essential points of the current analytical results and to motivate future analysis.

Local well-posedness of a nonlinear Fokker–Planck model

“…We showed local existence and uniqueness of the solution of the Fokker-Planck system. Large time asymptotic analysis of the proposed Fokker-Planck model, as well as numerical simulations of the system will be presented in a forthcoming paper [17]. As a part of our future research, we will further extend such Fokker-Planck systems to the modeling of the evolution of the grain boundary network that undergoes disappearance/critical events, e.g.…”

“…subject to the natural boundary condition, (a(x, t)f [17]. Let us briefly review the derivation [17].…”

“…subject to the natural boundary condition, (a(x, t)f [17]. Let us briefly review the derivation [17]. Indeed, by (2.8) and using the integration by parts, the rate of change of the free energy…”

“…As discussed above, such systems may arise in the grain growth modeling, e.g. [17,18]. In particular, the temperature, in terms of D in this context, will account for some information of the under-resolved mechanisms in the systems, such as critical events/disappearance events (e.g.…”