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

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.