Stochastic complex Ginzburg-Landau equation on compact surfaces

TL;DR

This paper investigates the stochastic complex Ginzburg-Landau equation on compact surfaces with magnetic effects, establishing local and global well-posedness results after appropriate renormalization and in specific regimes.

Contribution

It introduces a renormalization approach for the stochastic complex Ginzburg-Landau equation on compact surfaces and proves well-posedness results in both local and global settings.

Findings

Local well-posedness of the renormalized SCGL

Global well-posedness in the defocusing regime

Applicability to equations with magnetic Laplacian

Abstract

We study a stochastic complex Ginzburg-Landau equation (SCGL) on compact surfaces with magnetic Laplacian and polynomial nonlinearity, forced by a space-time white noise. After renormalizing the equation in a suitable manner, we show that the dynamics is locally well-posed. Moreover, we prove deterministic global well-posedness for the defocusing SCGL in the weakly dispersive regime.