Random attractor for the 2D stochastic nematic liquid crystals flows

Abstract

We consider the long-time behavior for stochastic 2D nematic liquid crystals flows with the velocity field perturbed by an additive noise. The presence of the noises destroys the basic balance law of the nematic liquid crystals flows, so we can not follow the standard argument to obtain uniform a priori estimates for the stochastic flow even in the weak solution space under non-periodic boundary conditions. To overcome the difficulty we use a new technique some kind of logarithmic energy estimates to obtain the uniform estimates which improve the previous result for the orientation field that grows exponentially w.r.t.time t. Considering the existence of random attractor, the common method is to derive uniform a priori estimates in functional space which is more regular than the solution space. We can follow the common method to prove the existence of random attractor in the weak solution space. However, if we consider the existence of random attractor in the strong solution space, it is very difficult and very complicated for such highly nonlinear stochastic system with no basic balance law and non-periodic boundary conditions. Here, we use a compactness arguments of the stochastic flow and regularity of the solutions to the stochastic model to obtain the existence of the random attractor in the strong solution space, which implies the support of the invariant measure lies in a more regular space. As far as we know, it is the first article to attack the long-time behavior of stochastic nematic liquid crystals.

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

“…Using the estimates in [1] or [8], we obtain the following proposition, Proposition 3.1. For v0 ∈ V, d0 ∈ H 2 and ω ∈ Ω. Denote by (u(t, v0, ω), d(t, d0, ω)) the unique solution to (3.3) on [0, T ].…”

“…The global well-posedness for the strong solution of (3.1) has been studied in [2] and [8], and it is known that under the condition (2.1), for any Let Z(t) be the unique solution of the stochastic equation:…”

“…Proof. Let (u(t, v0), d(t, d0)) t∈[0,T ] represent the unique strong solution to (3.3), see [8], where (u(t, v0), d(t, d0)) is shown to be Lipschitz continuous with respect to (v0, d0) in V × H 2 .…”

“…To see the compactness of the Fréchet derivative, we can follow the method in [8] and use the Aubin-Lions Lemma as well as the regularity of solutions. One can also adopt the method in Theorem 3.1 of [13] to show the compactness of (Dv(t, v0, ω), Dd(t, d0, ω))…”

Large Deviations for Stochastic Nematic Liquid Crystals Driven by Multiplicative Gaussian Noise

“…The unpublished paper [7] proved the existence and uniqueness of mxaimal local strong solution to the system (1.5)-(1.7) with a bounded nonlinear term f (n) = 1 |n|≤1 (1 − |n| 2 )n. The paper [6] deals only with weak (both in PDEs and stochastic calculus sense) solutions and the maximum principle. Some of the results in [6] and the current paper have already been used in several papers such as [9], [10], [65], [30], [29] and [64]. Very recently we have become aware of a recent paper by Feireisl and Petcu [26], in which they proved the existence of a dissipative martingale, as well as the existence of a local strong solution and weak-strong uniqueness of the solution of the stochastic Navier-Stokes Allen-Cahn Equations.…”