Asymptotic behavior of two-dimensional stochastic nematic liquid crystal flows with multiplicative noise

Abstract

Under non-periodic boundary conditions, we consider the long-term behavior of stochastic two-dimensional (2D) nematic liquid crystal flows with multiplicative noise. The structure of the corresponding stochastic model is different from that of the deterministic model. Noise destroys the basic balance law for the nematic liquid crystal flows, so we cannot employ the standard argument to obtain uniform a priori estimates of the solutions to this stochastic model. To overcome this problem, we use logarithmic energy estimates and the Itô formula in a Banach space to obtain uniform estimates that improve the previous result where the orientation field grows exponentially w.r.t. time t. In order to study the existence of a random attractor, we need to show that the solution is a stochastic flow. However, this is not obvious because of the particularly multiplicative noise in the orientation field. To show the flow property of the orientation field, we construct linear stochastic partial differential equations where the solutions are stochastic flows. After determining the relationship between these linear equations and the orientation field equation, we prove that each component of the orientation field is indeed a stochastic flow. The global well-posedness of the stochastic 2D nematic liquid crystal flows is only established for weak solutions, so the existence of the random attractor should be demonstrated in the weak solution space. Thus, using the standard method, we need to derive uniform a priori estimates in the strong solution space. However, the standard method fails because of the ill-posedness of the strong solution. We show that by proving the compactness property of the stochastic flow and using the regularity of the solutions, we can construct a compact absorbing ball in the weak solution space, which implies the existence of the random attractor. Finally, we show the existence of the invariant measure for the corresponding stochastic model, which is the first result for the long-term behavior of stochastic nematic liquid crystals under Dirichlet boundary conditions for the velocity field and Neumann boundary conditions for the orientation field.