Stochastic Nonparabolic dissipative systems modeling the flow of Liquid Crystals: Strong solution

Abstract

In this paper we present some mathematical results obtained from the analysis of a stochastic evolution equation which basically describes a Ginzburg-Landau approximation of the system governing the nematic liquid crystals under the influence of fluctuating eternal forces. We mainly prove the existence and uniqueness of local a maximal nd global strong solution to the problem. Here strong solution is understood in the sense of stochastic calculus and PDEs. By a fixed point argument we firstly prove a general result which enables us to establish the existence of local and maximal solution to an abstract nonlinear stochastic evolution equations. Secondly, we show that our problem falls within the previous general framework. Therefore we are able to establish the existence and uniqueness of local and maximal strong solution for both 2D and 3D case. In the 2D case we prove nonexplosion of the maximal solution by a method based on a choice of an appropriate energy functionals. Thus the existence of a unique global strong solution in the 2D case.