Some results on the penalised nematic liquid crystals driven by multiplicative noise: weak solution and maximum principle

Abstract

Abstract In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis and PDEs). Secondly, we show the pathwise uniqueness of the solution in a 2D domain. In contrast to several works in the deterministic setting we replace the Ginzburg–Landau function $$\mathbb {1}_{|{\mathbf {n}}|\le 1}(|{\mathbf {n}}|^2-1){\mathbf {n}}$$ 1 | n | ≤ 1 ( | n | 2 - 1 ) n by an appropriate polynomial $$f({\mathbf {n}})$$ f ( n ) and we give sufficient conditions on the polynomial f for these two results to hold. Our third result is a maximum principle type theorem. More precisely, if we consider $$f({\mathbf {n}})=\mathbb {1}_{|d|\le 1}(|{\mathbf {n}}|^2-1){\mathbf {n}}$$ f ( n ) = 1 | d | ≤ 1 ( | n | 2 - 1 ) n and if the initial condition $${\mathbf {n}}_0$$ n 0 satisfies $$|{\mathbf {n}}_0|\le 1$$ | n 0 | ≤ 1 , then the solution $${\mathbf {n}}$$ n also remains in the unit ball.