Existence and uniqueness of weak solutions to Ginzburg–Landau equation with external noise and stochastic perturbation

In this paper, the Ginzburg–Landau and fractional Ginzburg–Landau equations with external noise and stochastic perturbations are studied, and we obtain the global existence and uniqueness of weak L 1 -solutions. Our proof is supported by the generalized Itô formula and a careful analysis of the nonlinear transport equation. After founding the well-posedness for the nonlinear transport problem in the class of L p ( 1 < p ≤ ∞ ), we also gain the well-posedness of the Ginzburg–Landau and fractional Ginzburg–Landau equations in the same spaces, correspondingly.