Existence and Uniqueness of Weak Solutions for the Stochastic Fractional Ginzburg-Landau Equation

This paper investigates the existence and uniqueness of weak solutions for a stochastic Ginzburg-Landau equation involving the fractional Laplacian. The primary focus is on establishing a rigorous mathematical framework to handle the coexistence of the nonlocal fractional Laplacian and stochastic perturbations. By employing the Galerkin method, we establish that the initial-boundary value problem admits a unique global weak solution for any \( L^{2}_{a} \) initial value. This study utilizes the properties of the fractional Laplacian and fractional Sobolev spaces to provide a rigorous proof of the existence and uniqueness theorem. These results extend the analysis of Ginzburg-Landau equations to models incorporating stochastic terms and fractional Laplacian.