Global solutions for a nonlocal Ginzberg-Landau equation and a nonlocal Fokker-Plank equation

TL;DR

This paper establishes the existence and uniqueness of global solutions for nonlocal Ginzberg-Landau and Fokker-Planck equations using semigroup and viscosity methods, respectively, highlighting differences in solution regularity.

Contribution

It provides the first rigorous analysis of global solutions for these nonlocal equations with novel application of semigroup and viscosity methods.

Findings

Unique global solutions for nonlocal Ginzberg-Landau equation

Existence of solutions for nonlocal Fokker-Planck equation with weaker regularity

Differentiation in solution regularity due to drift term

Abstract

This work is devoted to the study of a nonlocal Ginzberg-Landau equation by the semigroup method and a nonlocal Fokker-Plank equation by the viscosity vanishing method. For the nonlocal Ginzberg-Landau equation, there exists a unique global solution in the set $C^0(\mathbb{R}^+,\,H_0^{\frac{\alpha}{2}}(D))\cap L_{loc}(\mathbb{R}^+,\,H_0^{\alpha}(D))$, for $\alpha\in (0,\,2)$. For the nonlocal Fokker-Plank equation, the regularity of the solution is weaker than that of the nonlocal Ginzberg-Landau equation due to the drift term.