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The flow of nematic liquid crystals can be described by a highly nonlinearstochastic hydrodynamical model, thus is often influenced by randomfluctuations, such as uncertainty in specifying initial conditions and boundaryconditions. In this article, we consider the $2$-D stochastic nematic liquidcrystals with the velocity field perturbed by affine-linear multiplicativewhite noise, with random initial data and random boundary conditions. Our mainobjective is to establish the global well-posedness of the stochastic equationsunder certain sufficient Malliavin regularity of the initial conditions and theboundary conditions. The Malliavin calculus techniques play important roles inproving the global existence of the solutions to the stochastic nematic liquidcrystal models with random initial and random boundary conditions. It should bepointed out that the global well-posedness is also true when the stochasticsystem is perturbed by the noise on the boundary.

Global well-posedness of stochastic nematic liquid crystals with random initial and random boundary conditions driven by multiplicative noise