Long-time asymptotic behavior of nonlinear Fokker-Planck type equations with periodic boundary conditions
Yekaterina Epshteyn, Chun Liu, Masashi Mizuno
TL;DR
The paper studies long-time behavior of nonlinear Fokker-Planck equations with non-homogeneous diffusion $D(x)$ and mobility $\pi(x,t)$ in a bounded domain with periodic boundaries, motivated by grain boundary dynamics in non-isothermal settings. It adopts an energetic-variational framework, employing the velocity field $\mathbf{u}=-\frac{1}{\pi}\nabla\left(D\log f+\phi\right)$ and the free energy $F[f]=\int_\Omega\left(D f(\log f-1)+f\phi\right)\,dx$, to derive higher-order decay properties of the system. The authors establish exponential decay of the dissipation (and of $-\frac{d}{dt}F[f](t)$) under regime-specific conditions: homogeneous diffusion, inhomogeneous diffusion, and the challenging case of variable mobility, with large $D_{\min}$ and controlled mobility derivatives. The results extend traditional entropy methods to bounded periodic domains with non-convex potentials, providing rigorous convergence guarantees that corroborate numerical observations in grain-boundary dynamics.
Abstract
In this paper, we study the asymptotic behavior of a class of nonlinear Fokker-Planck type equations in a bounded domain with periodic boundary conditions. The system is motivated by our study of grain boundary dynamics, especially under the non-isothermal environments. To obtain the long time behavior of the solutions, in particular, the exponential decay, the kinematic structures of the systems are investigated using novel reinterpretation of the classical entropy method.