The role of a strong confining potential in a nonlinear Fokker-Planck equation

TL;DR

This paper demonstrates how solutions of nonlinear nonlocal Fokker-Planck equations in bounded domains can be approximated by problems with strong confining potentials in the whole space, using energy and entropy estimates.

Contribution

It introduces two approaches for approximating bounded domain solutions via strong confining potentials, extending to degenerate diffusion cases.

Findings

Weak solutions in bounded domains are limits of solutions with increasing confining potentials.

Both energy and free energy methods effectively establish the approximation.

The free energy approach applies to degenerate diffusion scenarios.

Abstract

We show that solutions of nonlinear nonlocal Fokker--Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined in the whole space. Two different approaches are analyzed, making crucial use of uniform estimates for $L^2$ energy functionals and free energy (or entropy) functionals respectively. In both cases, we prove that the weak formulation of the problem in a bounded domain can be obtained as the weak formulation of a limit problem in the whole space involving a suitably chosen sequence of large confining potentials. The free energy approach extends to the case degenerate diffusion.