On strong solution to the 2D stochastic Ericksen-Leslie system: A Ginzburg-Landau approximation approach

Abstract

In this manuscript, we consider a highly nonlinear and constrained stochastic PDEs modelling the dynamics of 2-dimensional nematic liquid crystals under random perturbation. This system of SPDEs is also known as the stochastic Ericksen-Leslie equations (SELEs). We discuss the existence of local strong solution to the stochastic Ericksen-Leslie equations. In particular, we study the convergence the stochastic Ginzburg-Landau approximation of SELEs, and prove that the SELEs with initial data in H 1 ×H 2 has at least a martingale, local solution which is strong in PDEs sense.

Global weak solutions to the Stochastic Ericksen--Leslie equations in dimension two

“…Analogously, Bouard-Hocquet-Prohl obtained the Struwe-like global solution to (1.1) in [2] by a bootstrap argument together with Gyöngy-Krylov L p estimates [14]. Very recently, Brzeźniak, Deugoué, and Razafimandimby in [3] proved the existence of short time strong solutions to the simplified stochastic Ericksen-Leslie system. The main goal of this paper is to obtain a global weak solution to (1.1) by extending the compactness argument from [10] into the stochastic setting.…”