Random attractor for the 2D stochastic nematic liquid crystals flows with multiplicative noise

Under non-periodic boundary conditions, we consider the long-time behavior for stochastic 2D nematic liquid crystals flows with velocity and orientations perturbed by additive noise and multiplicative noise respectively. It is the first result for the long-time behavior of stochastic nematic liquid crystals under Dirichlet boundary condition for velocity field and Neumann boundary condition for orientation field.


Introduction
The paper is concerned with the following stochastic hydrodynamical model for the flow of nematic liquid crystals in D × R + , where D ⊂ R 2 is a bounded domain with smooth boundary Γ.
The unknowns for the 2D stochastic hydrodynamical model are the fluid velocity field v = (v 1 , v 2 ) ∈ R 2 , the averaged macroscopic/continuum molecular orientations d = (d 1 , d 2 , d 3 ) ∈ R 3 and the scalar function p(x, t) representing the pressure (including both the hydrostatic and the induced elastic part from the orientation field). The positive constants ν, λ and γ stand for viscosity, the competition between kinetic energy and potential energy, and macroscopic elastic relaxation time (Debroah number) for the molecular orientation field. W 1 is a standard Wiener process in H defined below and has the form of where ((B i (t)) t∈R ) i∈N be a sequence of one-dimensional, independent, identically distributed Brownian motions defined on the complete probability space (Ω, F, P), (e i ) i∈N is an orthonormal basis in H, (λ i ) i∈N is a convergent sequence of positive numbers which assure W 1 is a standard Wiener process in H. W 2 is a standard real-valued Brownian motion with (d × h) •Ẇ 2 understood in the stratonovich sense. h = (h 1 , h 2 , h 3 ) ∈ R 3 , h i is constant, i = 1, 2, 3. Here f : R 3 → R 3 is a general polynomial function whose details will be given later. The symbol ∇d⊙ ∇d denote the 2× 2 matrix whose (i, j)-th entry is given by In this paper, we consider the following initial boundary conditions for the stochastic nematic liquid crystals equations. Boundary conditions v(t, x) = 0, ∂d(x, t) ∂n = 0, for (x, t) ∈ Γ × R + . (1.4) Initial conditions where n is the outward unite normal vector to Γ. In [18], F.H. Lin proposed a corresponding deterministic model of (1.1) − (1.3) as a simplified system of the original Ericksen-Leslie system (see [10,17]). By Ericksen-Leslie's hydrodynamical theory of the liquid crystals, the simplified system describing the orientation as well as the macroscopic motion reads as follows dv + [(v · ∇)v − µ∆v + ∇p]dt = λ∇ · (∇d ⊙ ∇d)dt, (1.6) ∇ · v(t) = 0, (1.7) ∂ t d + [(v · ∇)d] = γ(∆d(t) + |∇d| 2 d), |d| = 1. (1.8) In order to avoid the nonlinear gradient in (1.8), usually one uses the Ginzburg-Landau approximation to relax the constraint d = 1. The corresponding approximate energy is (1.9) where η is a positive constant. Then one arrives at the approximation system (1.6) − (1.8) with f (d) and F (d) given by f (d) = 1 η 2 (|d| 2 − 1)d and F (d) = 1 4η 2 (|d| 2 − 1) 2 .
(1. 10) In this work, we consider a more general polynomial function f (d) which contains as a special case the (1.10). We define a functionf : [0, ∞) → R bỹ a k x k , x ∈ R + , (1.11) where a N > 0 and a k ∈ R, k = 0, 1, 2, ..., N − 1. Let f : R 3 → R 3 given by f (d) =f (|d| 2 )d. (1.12) Denote by F : R 3 → R the Fréchet differentiable map such that for any d ∈ R 3 and ξ ∈ R 3 SetF to be an antiderivative off such thatF (0) = 0. Theñ The Ericksen-Leslie system is well suited for describing many special flows for the materials, especially for those with small molecules, and is widely accepted in the engineering and mathematical communities studying liquid crystals. System (1.1)-(1. 3) with f (d) given by (1.10) can be possibly viewed as the simplest mathematical model, which keeps the most important mathematical structure as well as most of the essential difficulties of the original Ericksen-Leslie system (see [19]). This deterministic system with Dirichlet boundary conditions has been studied in a series of work not only theoretically (see [19], [20]) but also numerically (see [24], [25]).
The introduction of stochastic processes in nematic liquid crystals flows is aimed at accounting for a number of uncertainties and errors: (1)The state of the nematic liquid crystals is strongly dependent on the state of the environment. In natural systems external noise is often quite large. At an instability point the systern is sensitive even to infinitesimally small perturbations and the role of external noise has to be investigated in the vicinity of a transition point. And experimental investigation also showed that, in the case of electrohydrodynamic instabilities, the average value of the voltage necessary for the transition is shifted to higher and higher values as the intensity of the external noise is increased i.e., the amplitude of the voltage fluctuations, is increased. Further study showed that the average voltage necessary to induce the transition to turbulent behavior, increases with the variance of the voltage fluctuations (see [14] ). For more details one can also refer to [3,4,5,15,30,31].
(2)Rheological predictions of the behavior of complex fluids like these, often start with the derivation of macroscopic, approximate equations for quantities of interest using various closure approximations. The difficulty in obtaining accurate closures has motivated the extensive, in recent years, use of direct simulations, either of the PDE governing the orientation distribution function, or of the equivalent stochastic differential equation, via Brownian dynamics simulations. The latter have the advantage that they are amenable to use with models with many internal degrees of freedom (as opposed to the PDE approach in which the curse of dimensionality precludes realistic computation). For more details one can see [11,16,23,28,29].
Despite the developments in the deterministic case, the theory for the stochastic nematic liquid crystals remains underdeveloped. To the best of our knowledge, there are few works on the stochastic nematic liquid crystals. In the papers [3,4], Z.Brzezniak, E.Hausenblas and P.Razafimandimby have considered the model perturbed by multiplicative Gaussian noise and have proved the global well-posedness for the weak solution and strong solution in 2-D case. When the noise is jump and the dimension is two, Z.Brzezniak, U.Manna and A.A. Panda in [5] obtained the same result as the case of Gaussian noise. A weak martingale solution is also established for the three dimensional stochastic nematic liquid crystals with jump noise in [5].
One natural problem arising from this global existence result is the dynamical behavior of the 2D stochastic system.
Under the periodic boundary conditions or the assumption d(t, x) = d 0 (x) for (x, t) ∈ Γ × R + , the existence of global attractor is established for the deterministic model(see [13,27]). However, there is no published result about the existence of random attractor for the stochastic nematic liquid crystals flows. One reason is the absence of the basic balance law which results in the failure of applying the method of deterministic model to the stochastic model. The other important reason is due to the different boundary conditions between the deterministic model and stochastic model. As we see if the boundary condition is given by (1.4), the following equality is not ture where , and | · | 2 denote the inner product and norm in (L 2 (D)) 3 . Therefore, we can not obtain the global existence of the strong solution in partial differential equations sense.
In this article, we firstly improve the bounds for the solutions to (1.1) − (1.5). These bounds are uniform with respect to present time and initial time (see Lemma 3.1). These estimates of the uniform boundedness improve previous bounds obtained in [3,4,5], in which the bounds of d grow exponentially with respect to present time or initial time. In obtaining these time-uniform a priori estimates for orientation field d (for example in space (L 2 (D)) 3 ), the power of d from the nonlinear f (d) will be much bigger (see (1.12)) than two. Using the Gronwall inequality (a standard argument) we will obtain that the solutions have exponential growth which is not sufficient to ensure the existence of random attractor. To overcome the difficulty, our idea is that after applying Itô formula in Banach space (L 4N +2 (D)) 3 , N > 1, to d, we try to take advantage of the property of the logarithmic function to reduce the power from nonlinear term. Roughly speaking, for a positive f (t), t ∈ R + , if we want to consider it's uniform boundedness with respect to time t, we just need to estimate ln(1 + f (t)). If ln(1 + f (t)) is uniformly bounded with respect to t, so is f (t). Using this new technique we obtain the uniform estimates for orientation field d ( see Lemma 3.1) which opens a way to study the long-time behavior of stochastic nematic liquid crystals.
To show the existence of random attractor, another problem needed to be addressed is to prove the solution (v, d) to (1.1) − (1.5) is indeed a stochastic flow. The difficulty lies in (1.3), we should show d to (1.3) is a stochastic flow. But for this equation, we can not follow the method in [8] of constructing a equivalent vector valued partial differential equation with random coefficients to prove that the stochastic orientation field is a stochastic flow. To overcome the difficulty we construct a linear stochastic partial differential equations (SPDE) with linear Stratonovich multiplicative noise whose scalar valued solution is a stochastic flow. Then for different h, we will obtain different linear and scalar valued SPDEs whose solutions are stochastic flow. Taking advantage of the relationship between these linear SPDEs and (1.3), we can prove one component of the solution d to (1.3) is indeed a stochastic flow. Then repeating the arguments, we can infer that each component of d is a stochastic flow which implies the flow property of stochastic field d. For the detail we can refer to Proposition 3.2 and Remark 3.3.
Our main goal of this article is to show the existence of random attractor in the solution space H × H 1 . As we know, the sufficient condition for ensuring the existence of random attractor in H × H 1 is to obtain an absorbing ball which is compact in H × H 1 . The common method is to derive uniform a priori estimates in the functional space V × H 2 which is the strong solution space. However, the global existence of strong solution is unavailable due to the Neumann boundary condition. Here, we use a compactness arguments of the stochastic flow and regularity of the solutions to construct a compact absorbing ball in the function space H × H 1 . We complete the proof of the existence of random attractor by four steps. Firstly, using the Lemma 3.1, we obtain the absorbing ball in the weak solution H × H 1 (see Proposition 3.1). Secondly, we will verify the two a priori estimates of Aubin-Lions compact lemma to obtain a convergent subsequence of (v, d) which converges almost everywhere with respect to time t ∈ [s, T ], −∞ < s < T < ∞ (see Proposition 3.3 and Proposition 3.4). Thirdly, in order to show the solution operator is a stochastic dynamical system we show that v ∈ C([0, T ]; H) and d ∈ C([0, T ]; H 1 ), which improves the regularity of the solution (v, d) obtained in [3]. For the details we can refer to Corollary 3.1. Using the regularity of the solutions to (1.1) − (1.5) and the Aubin-Lions Lemma we prove in Proposition 3.5 that the solution operators are almost surely compact in H × H 1 for all (t, s) satisfying −∞ < s < t < ∞. Finally, in Proposition 3.6 using the compact solution operator to act on the the absorbing ball yields a new set which is compact and absorbing in H × H 1 . The existence of the random attractor in the weak solution space H × H 1 follows directly from Proposition 3.6.
This article illustrates some advantage of our method over the common method. For the present model, following the common method to prove the existence of random attractor, we need to obtain uniform a priori estimates in a function space V × H 2 . As we see it is very difficult, i.e., the uniform a priori estimates for (1.1) − (1.5) in more regular function space than the solution space is not available. Our method here avoid doing estimates in function spaces V×H 2 , but prove the compact absorbing ball in H × H 1 indeed exists .
The remaining of this paper is organized as follows. In section 2, we state some preliminaries and recall some results. The existence of random attractor is presented in section 3. As usual, constants C may change from one line to the next, unless, we give a special declaration ; we denote by C(a) a constant which depends on some parameter a.

Preliminaries
For 1 ≤ p ≤ ∞, let L p (D) be the usual Lebesgue spaces with the norm | · | p . For a positive integer m, we denote by (H m,p (D), · m,p ) the usual Sobolev spaces, see( [1]). When p = 2, we denote by (H m (D), · m ) with inner product , H m . Let We denote by H, V and H 2 be the closure spaces of V in (L 2 (D)) 2 , (H 1 (D)) 2 and (H 2 (D)) 2 respectively. And set |·| 2 and , to be the norm and inner product of H respectively. The notation , is also used to denote the inner product in (L 2 (D)) 2 . By the Poincaré inequality, there exists a constant c such that for any v ∈ V we have v 1 ≤ c|∇v| 2 . Without confusion, we let · 1 and , V stand for the norm and the inner product in V respectively, where , V is defined by Denote by V ′ the dual space of V. And define the linear operator A 1 : V → V ′ , as the following: Since the operator A 1 is positive selfadjoint with compact resolvent, by the classical spectral theorems there exists a sequence {α j } j∈N of eigenvalues of A 1 such that (2.14) For arbitrary constant T > 0 and j ∈ N, we define and Obviously, For k ∈ N and k > j, In view of an infinite dimensional version of Burkholder-Davis-Gundy type of inequality for stochastic convolutions (see Theorem 1.2.6 in [9,22] and references therein), we have Then, similarly, we define the linear operator A 2 : are self-adjoint compact operators in H and H respectively, thanks to the classic spectral theory, we can define the power A s i for any s ∈ R. Then D( Similarly, To prove the existence of random attractor for stochastic liquid crystals flows, we need the following result concerning global well-posedness of (1.1) − (1.5). For the proof, one can follow the argument as in [3] or [5] with minor revisions. .
But this kind of regularity can not make the solution operator be a stochastic dynamical system. In In the following, we recall the notations and results in stochastic dynamical systems which will be use to prove the main results of this article.
Let (X, d) be a polish space and (Ω,F,P) be a probability space, whereΩ is the two -sided Wiener space C 0 (R; X) of continuous functions with values in X, equal to 0 at t = 0. We consider a family of mappings S(t, s; ω) : X → X, −∞ < s ≤ t < ∞, parametrized by ω ∈Ω, satisfying for P-a.e. ω the following properties (i)-(iv): (i) S(t, r; ω)S(r, s; ω)x = S(t, s; ω)x for all s ≤ r ≤ t and x ∈ X; (ii) S(t, s; ω) is continuous in X, for all s ≤ t; (iii) for all s < t and x ∈ X, the mapping ω → S(t, s; ω)x is measurable from (Ω,F ) to (X, B(X)) where B(X) is the Borel-σ-algebra of X; (iv) for all t, x ∈ X, the mapping s → S(t, s; ω) is right continuous at any point. A set valued map K :Ω → 2 X taking values in the closed subsets of X is said to be measurable if for each x ∈ X the map ω → d Moreover, if for all bounded sets B ⊂ X, there exists t B (ω) such that for all s ≤ t B (ω) we say K(t, ω) is an absorbing set at time t. Let {ϑ t :Ω →Ω}, t ∈ T, T = R, be a family of measure preserving transformations of the probability space (Ω,F ,P) such that for all s < t and ω ∈Ω (a) (t, ω) → ϑ t ω is measurable; (c) S(t, s; ω)x = S(t − s, 0; ϑ s ω)x. Thus (ϑ t ) t∈T is a flow, and ((Ω,F,P), (ϑ t ) t∈T ) is a measurable dynamical system.  Furthermore, if there exists a compact attracting set K(t, ω) at time t, it is not difficult to check that A(B, t, ω) is a nonempty compact subset of X and A(B, t, ω) ⊂ K(t, ω).
To prove the existence of the random attractor, we will use the following sufficient condition given in [7]. For the convenience of reference, we cite it here.
Theorem 2.2 Let (S(t, s; ω)) t≥s,ω∈Ω be a stochastic dynamical system satisfying (i), (ii), (iii) and (iv). Assume that there exists a group ϑ t , t ∈ R, of measure preserving mappings such that condition (c) holds and that, forP-a.e. ω, there exists a compact attracting set K(ω) at time 0. ForP-a.e. ω, we set where the union is taken over all the bounded subsets of X. Then we have forP-a.e. ω ∈Ω.
(1) A(ω) is a nonempty compact subset of X, and if X is connected, it is a connected subset of K(ω).
Moreover, it is the minimal closed set with this property: ifÃ(ϑ t ω) is a closed attracting set, then A(ϑ t ω) ⊂Ã(ϑ t ω).  In this article, to prove our main result we will use the compactness arguments of solution operator and the regularity of the solution to (1.1) − (1.5) which rely on the two lemmas below. The first lemma is Aubin-Lions Lemma whose proof can be founded in [2,21]. The following lemma, a special case of a general result of Lions and Magenes [26], will help us to show the continuity of the solution to stochastic nematic liquid crystals with respect to time.

Random attractor for the weak solution
One of our main results in this article is to prove: Proof. The results of this theorem follows directly from Proposition 3.6 and Theorem 2.2. The rest of this section is to find a compact absorbing ball for (1.1) − (1.5) in H × H 1 . We will achieve our goal by six steps. In subsection 3.1, we use a new technique logarithmic energy estimates combined with Itô formula to obtain the uniform a priori estimates in (L 4N +2 (D)) 3 which is very important to study the long-time behavior of the stochastic nematic liquid system (see Lemma 3.1 and the proof of Proposition 3.1). Then in subsection 3.2, we obtain the absorbing ball for the solution to (1.1) − (1.5) in the space H × H 1 in Proposition 3.1. As the third step, in subsection 3.3, we use a new technique to prove the solution is indeed a stochastic flow in Proposition 3.2. In the next subsection, by Proposition 3.3 and Proposition 3.4 we verified two a priori estimates of the Aubin-Lions lemma which is used to obtain a convergent subsequence of the solutions to (1.1) − (1.5). In subsection 3.5, taking advantage of the convergent subsequence and the continuity of the solutions with respect to initial data in H × H 1 , we prove the solution operator S(t, s; ω) is almost surely compact from H × H 1 to H × H 1 for all s, t ∈ R, s < t (see P roposition 3.5). Finally, using the existence of absorbing ball and compactness of the solution operator, we construct a compact absorbing ball in Proposition 3.6.
To study the long time behavior of (1.1)− (1.5), we introduce a modified stochastic convolution. Let t ∈ R and β ∈ R + . For simplicity, we still define Then by (2.15), we have z(ω) ∈ C([0, T ]; H 2 ), P − a.e. and satisfies the linear equation du + ((u + z) · ∇(u + z) + ∇p − ∆u)dt + ∇ · (∇d ⊙ ∇d)dt = βz, These uniform bounds also allow us to obtain the absorbing balls for the solution in various function spaces(see (3.36)). The radii of these absorbing balls are independent of the initial data. This opens the way for finding the random attractor in the weak solution space H × H 1 . Maybe this lemma is a basic result to study the long-time behavior of stochastic nematic liquid crystals i.e. the existence of random attractor and ergodicity for this stochastic system.
Proof. Since |d| 4N +2 4N +2 is twice Fréchet differentiable for d ∈ H 2 , then the first and second derivatives of |d| 4N +2 4N +2 are given by By elementary calculus, we have Then, applying Itô formula to |d| 4N +2 4N +2 we have which implies that By Young's inequality, for small positive constant ε there exists a positive constant c such that Therefore, By (3.24) we have Let y(t) = ln(|d(t)| 4N +2 4N +2 + 1). Then multiplying e ct on both sides yields which implies Proof. The first order and second order Fréchet derivative of |∇d| 2 2 + D F (|d(x)| 2 )dx are given by and Since ∆d−f (d), d×h dW 2 = 0 and d ⊥ d×h, applying Itô formula to |∇d| 2 2 + DF (|d(x)| 2 )dx, d ∈ H 1 , yields Taking inner product of (3.16) with u in H yields, Since by integration by parts and boundary condition (3.16) , combining (3.28) and (3.29) together yields 1 2 for all time t ∈ R and initial time t 0 (≤ t). Since by the Poincaré inequality and the Minkovski inequality Since, the process z(t) is stationary and ergodic and z(t) 1 has polynomial growth when t → −∞, following the classic arguments (see [7,9] and other references), (3.37) gives us the desired uniform estimate which yields an absorbing ball for (u, d) in H × H 1 . Following the standard arguments, we can also show that for any constant r and t there exists t t−r (|∇u(s)| 2 2 + |∆d(s)| 2 2 )ds is uniformly bounded with respect to initial time t 0 ≤ t − r.
Proof. To show (v, d) is a stochastic flow, it is sufficient to show d is a stochastic flow.
Obviously,d is a stochastic flow (see [8]). Leth = (h 1 , −h 2 , −h 3 ). Define α 1 = e W 2 (t)h and d 1 = α 1 · d. Then following the above steps we will show d 1 is also a stochastic flow. Since d = (d 1 , d 2 , d 3 ) and Similarly, we can show d 2 and d 3 is a stochastic flow. Then the conclusions of the proposition follows.

Two a priori estimates for the Aubin − Lions Lemma.
To establish that the solution operator to (1.1)−(1.5) is compact in H×H 1 , the first step is to use the Aubin-Lions lemma to obtain a convergent subsequence of (v, d) or equivalently a convergent subsequence of (u, d) which converges almost everywhere with respect to time t ∈ [s, T ], s, T ∈ R and s < T, in H × H 1 . Then we have to verify the two a priori estimates of Aubin-Lions lemma.
Hence for any t ∈ (s, T ], {S(t, s; ω)(v 0,n , d 0,n )} n∈N contains a subsequence which is convergent in H × H 1 , which implies that for any fixed t ∈ (s, T ], ω ∈ Ω, S(t, s; ω) is a compact operator in H × H 1 .
3.6. The existence of compact absorbing ball in H × H 1 .
Proposition 3.6 There exists a compact absorbing ball at any time t ∈ R for the stochastic dynamical system (1.1) − (1.5) in H × H 1 .
Proof. Using the notations given in Proposition 3.5, for s < t, let B(t, ω) = S(t, s; ω)B(s, r(ω)) be the closed set of S(t, s; ω)B(s, r(ω)) in H×H 1 , where B(s, r(ω)) is the absorbing ball at time s with center 0 ∈ H × H 1 and radius r(ω). Then, by the above arguments, we know B(t, ω) is a random compact set in H × H 1 for each ω.  where B ∈ B(H × H 1 ) which is a Borel σ-algebra on H × H 1 . If the invariant measure for P t is unique, then the invariant Markov measure µ · for ϕ is unique and given by µ ω = lim t→∞ ϕ(t, ϑ −t ω)µ.