A note on the Stochastic Ericksen-Leslie equations for nematic liquid crystals

In this note we prove the existence and uniqueness of local maximal smooth solution of the stochastic simplified Ericksen-Leslie systems modelling the dynamics of nematic liquid crystals under stochastic perturbations.

1. Introduction. Liquid crystal, which is a state of matter that has properties between amorphous liquid and crystalline solid can be classified into two groups according to the form of their molecules. Liquid crystals with rod-shaped molecules are called calamitics while those with disc-like molecules are referred to discotics. In its turn, the calamitics can be divided into two phases: nematic and smectic. The nematic phase, referred to as nematic liquid crystal, is the simplest of liquid crystal phases. Nematic liquid crystals tend to align along a particular direction denoted by a unit vector d, called the optical director axis. Most of the interesting phenomenology of nematic liquid crystals are linked to the geometry and dynamics of this director. We refer to [10] and [15] for a comprehensive treatment of the physics of liquid crystals.
To model the hydrodynamics of nematic liquid crystals, most scientists use the continuum theory developed by Ericksen [17] and Leslie [33]. From this theory, F. Lin and C. Liu [34] derived the most basic and simplest form of the dynamical system describing the motion of nematic liquid crystals flowing in R d pd " 2, 3q. This system is given by v t`p v¨∇qv´∆v`∇p "´λ∇¨p∇d d ∇dq, (1.1a) ∇¨v " 0, (1.1b) d t`p v¨∇qd " γp∆d`|∇d| 2 dq, (1.1c) Here p : R d Ñ R, v : R d Ñ R d and d : R d Ñ R 3 represent the pressure, velocity of the fluid and the optical director, respectively. The symbol ∇d d ∇d stands for a square dˆd-matrix with entries given by , for any i, j " 1, . . . , d.
In this paper we consider the following system of stochastic partial differential equations (SPDEs) a separable Hilbert space K 1 , η is a one-dimensional standard Brownian motion, and Q is a nonlinear map satisfying several conditions specified later on.
Throughout this paper we assume that v, d, p, as well as h are 2π-periodic in the following sense: In what follows, when we refer to problem (1.2), we refer to the system of equations (1.2) with the boundary condition given in (1.3).
The system of SPDEs (1.2) describes the dynamics of nematic liquid crystal with a stochastic perturbation. Our investigation is motivated by the need for a mathematical analysis of the effect of the stochastic external perturbation on the dynamics of nematic liquid crystals. While the role of noise on the dynamics of d has been the subject of numerous theoretical and experimental studies in physics, see, for instance, [29,45,46], in which it is found that the time needed by the system to leave an unstable state diminishes in the presence of fluctuating magnetic fields, there are almost no rigorous mathematical results in his direction of research. The works [29,45,46] and the mathematical paper we cited earlier neglected either the effect of the velocity v or the stochastic external perturbation, although, de Gennes and Prost [15] noted that v plays an essential role in the dynamics of d. It is this gap in knowledge that is the motivation for our mathematical study. The current authors established in [7] some existence, uniqueness and a maximum principle results for the stochastic version of a Ginzburg-Landau approximation of the system (1.2) without the sphere condition (1.2e).
In this paper we study the local resolvability of problem (1.2). Our result can be summarized as follows. Given a number α ą d 2 and a square integrable H α solĤ α`1 -valued random variable pv 0 , d 0 q we can find a stopping timeτ 8 which can be approximated by an increasing sequence of stopping times pτ m q mPN and a unique local stochastic process pv, dq " pvptq, dptqq, 0 ď t ăτ 8 satisfying the following conditions Moreover, we established probabilistic lower bounds on the lifespanτ 8 of the local maximal solution.
These results extend to the stochastic case the local existence and uniqueness results for (1.1) obtained for the deterministic model by Wang et al. in [50]. Our proof consists of two steps. In the first one, we apply earlier results obtained in [7] to prove the existence and uniqueness of a maximal local solution satisfying the mild form of equations (1.2a)-(1.2d). In the second one, we prove that when properly localised the local solution preserves the sphere condition (1.2e).
The structure of the paper is as follows. In section 2 we present the main notation and standing assumptions we will be using in the whole paper. In section 3, we introduce the concept of a solution and state our main results. The proof of the main theorems are given in section 4 and section 5.

Functional spaces and hypotheses.
We begin by introducing the necessary definitions of functional spaces frequently used in this work. We denote by O the ddimensional torus d " 2, 3. Functions defined on O will be frequently identified with functions defined on the set r´π, πs d satisfying appropriate to their regularity periodic boundary conditions, for example, (1.3).
Throughout this paper we denote by L p pOq and W m,p pOq, p P r1, 8s, m P N, the Lebesgue and Sobolev spaces of real valued functions defined on O, see e.g. the monograph [48] by Temam (compare [3]). The corresponding spaces of R d (or some cases R 3 )-valued functions, will be denoted by the black-board fonts, e.g. the space L p pO, R d q will be denoted by L p pOq.
For non-integer r ą 0 the Sobolev spaces H r,p pOq and H r,p pOq are defined by using the complex interpolation method. We will also use the notation H r pOq :" W r,2 pOq. We simply skip the symbol of the torus O, when there is no risk of ambiguity. For instance we will write L p , resp. L p of W m,p instead of L p pOq, resp. L p pOq or W m,p pOq.
Given two Banach spaces K and H, we denote by L pK, Hq the space of bounded linear operators. For two Hilbert space K and H we denote by L 2 pK, Hq the Hilbert space of all Hilbert-Schmidt operators from K to H. For K " H we just write L pKq instead of L pK, Kq.
Following [48] we also introduce the following spaces In the above formula, the divergence is understood in the weak sense. Note that H 0 sol " H. In (1.2), it is convenient to eliminate the pressure p by applying the Helmholtz-Leray projector operator Π : L 2 Ñ H which projects into divergence free vectors and annihilates gradients. One of the remarkable properties of Π is that Π P L pH r , H r sol q, r ą 0, see [4]. We will frequently use this property without further notice.
Next, we define the Stokes operator, denoted by A, which is an unbounded linear operator on H, as follows.
It is well known that A is a strictly positive self-adjoint operator in H and that DpA 1{2 q " V. It is also true that A is a strictly positive self-adjoint operator in every space H r sol , r ą 0. We will also need a version of the Laplace operator acting on R 3 -valued functions defined on O, i.e.

"
DpA 2 q :" H 2 pO, R 3 q, It is well known that A 2 is a non-negative self-adjoint operator in L 2 pO, R 3 q. It is also true that A 2 is a non-negative self-adjoint operator in every space H r pO, R 3 q, r ą 0. It is well-known that´A (resp.´A 2 ) is the infinitesimal generator analytic C 0 -semigroup of contractions on H, resp. L 2 pO, R 3 q. These semigroups will be denoted by tSptq : t ě 0u and tTptq : t ě 0u. Moreover, for s 1 ą s there exists a constant M (depending on the difference s 1´s and p) such that we have (compare Lemma 1.2 in the Kato-Ponce's paper [31]) Let us note the following inequality involving fractional Sobolev norms.
Later on, we will state and prove few crucial properties of these nonlinear maps.
Let us fix d P H α`1 and set It is easy to see that G P L pH α`1 q. Let us note that the map G 2 , also an element of L pH α`1 q, is of the following form Let pΩ, F , Pq be a complete probability space equipped with a filtration F " tF t : t ě 0u satisfying the usual condition. LetW " pWptqq tě0 be a cylindrical Wiener process evolving on a separable Hilbert space K 1 such that it is formally written as a seriesW ptq " where pw k ptqq kPN,tě0 is a family of i.i.d. standard Brownian motions and tϕ k ; k P Nu is an orthonormal basis of K 1 . The above series does not converge in K 1 , but it does converges in a separable Hilbert spaceK 1 such that the embedding K 1 ĂK 1 is Hilbert-Schmidt. It is well-known also thatW has a modification, still denoted by W, whose trajectories are continuousK 1 -valued functions. Let η be a standard one dimensional Brownian motion and h be a smooth vector fields.
We now introduce the assumption on the coefficient Q of the noise.

Assumptions 1.
We fix α ą d{2 and we assume that Q : H α sol Ñ L 2 pK 1 , H α sol q is a globally Lipschitz map. In particular, there exists ℓ 0 ě 0 such that Hereafter we set (2.10) The stochastic equations for nematic liquid crystal (1.2) can be rewritten as a stochastic evolution equation in the space H α : where, for y " pv, dq P E α and k " pk 1 , k 2 q P K :" K 1ˆR , we have The process W is a cylindrical Wiener process on K such that for any t ě 0 Wptq "ˆW ptq ηptq˙, t ě 0.

Existence and uniqueness of local maximal solution.
We first recall several definitions and concepts which are given in the following notations/definitions and are borrowed from [5] or [32]. [32, p. 45]) For a probability space pΩ, F , Pq with a given right-continuous filtration F "`F t˘t ě0 , a stopping time τ is called accessible iff there exists an increasing sequence of stopping times τ n such that a.s. τ n ă τ and lim nÑ8 τ n " τ.
We now define some concepts of solution to (2.11), see [8,Def. 4.2] A local mild solution to problem (2.11) with initial condition yp0q " y 0 is a pair py, τq such that 1. τ is an accessible F-stopping time, 2. y : r0, τqˆΩ Ñ V α is an admissible process, 3. there exists an approximating sequence pτ m q mPN of finite F-stopping times such that τ m Õ τ a.s. and, for every m P N and t ě 0, we have for all x P O.
We also introduce the notion of maximal local solution and its lifespan. Having defined our solution concept, we can now state and prove the existence of a maximal local solution for our model. We also give a lower estimate and a characterisation of the local solution's lifespan.  We will show in the next theorem that the local solution from Theorem 3.6 satisfies (1.2e).

Theorem 3.7.
Assume that all the assumption of Theorem 3.6 are satisfied. Let y 0 " pv 0 , d 0 q P V α such that |d 0 pω, xq| 2 " 1 for all ω P Ω and all x P O. Let py; τq " ppv, dq; τq be a local solution to (2.11) and pτ m q mPN an increasing sequence of stopping times approximating τ. Then, for all t P p0, Ts P-a.s. |dpt^τ m , x, ωq| 2 " 1 for all x P O. Remark 3.8. We suspect that, if d P t2, 3u, α ą d{2, then under reasonable assumptions about the noise, there exists a local maximal solution for every initial data y 0 " pv 0 , d 0 q P H α´1 solˆH α . We also suspect that the existence of a local solution is mainly due to the mathematical analysis. We limited ourselves to the analysis of local solution as we were not able to derive proper estimates yielding global existence. We, however, have the conjecture that under smallness condition on the initial data one should be able to prove global existence of solution; this is the case for the deterministic model, see [50]. These questions will be investigated in subsequent papers.
The proofs of these two theorems are given in sections 4 and 5, respectively.

Proof of Theorem 3.6.
In order to prove the results in Theorem 3.6 we will use the general results proved in [7,Theorem 5.15 and 5.16]. For this purpose we establish several crucial estimates for the nonlinear terms in (1.2) in the following lemmata.
Lemma 4.1. Assume that α ą d{2. Then, there exist δ P r0, 1q and C ą 0 such that for any u P H α sol , v P H α`1 sol and d, m P H α`1 Proof of Lemma 4.1. Let u P H α sol , v P H α`1 sol and d, m P H α` 1 . In what follows we will denote by C various generic constants not depending neither on u, v, d nor m. By the inequality (2.5), we get Since α ą d{2, one can find a positive constant δ P p0, 1q such that α´δ ą d{2. Thus, by the Sobolev embedding H α´δ Ă L 8 and the Gagliardo-Nirenberg inequality we have from which we infer that The first estimate in our lemma easily follows from this last line and the fact that (as we are on the torus) the Leray-Helmhotz projection operator Π belongs to L pH α´1 , H α´1 sol q. We now prove the second estimate. As α ą d{2, H α sol is an algebra and we can easily infer that from which the second estimate in our lemma easily follows. Now we deal with third estimate where the nonlinear map M is involved. Since Using an argument similar to the proof of (4.2) yields (4.3).
We will also need the following lemma.
We now are ready to embark on the promised proof of Theorem 3.6.
Proof of Theorem 3.6. Since the maps M, B andB are bilinear, we infer from the Lemmata 4.1 and 4.2 that the nonlinear term F defined in (2.12) satisfies the following property: There exist two constants δ P p0, 1q and C ą 0 such that for any y 1 , y 2 P E α we have From the definition 2.9 of the map G and the assumption on h it follows that L P L pH α solˆH α`1 q from which, along with (4.6), we infer that F`L satisfies Assumption 2 of Theorem A.1 ( see also [7,Assumption 5.1]).
Because of Assumption 1 and the fact that G P L pH α`1 q it is clear that G satisfies Assumption 3 of Theorem A.1. Now, let X T be the Banach space with the norm defined by Sp¨´sqgpsqdWpsq, g P M 2 p0, T; L 2 pK, V α qq, is also bounded. From the observations above, Assumption 1 and the assumption on the initial data y 0 we infer that the problem (2.11) satisfies all the assumptions of Theorems A.1 and A.2 ( see also [7,Theorem 5.15 and 5.16]) from which we easily complete the proof of the Theorem 3.6.

Proof of Theorem 3.7.
In this section we give the proof of the sphere constraint.
Proof of Theorem 3.7. The proof will be divided into two steps.
Let tφ ℓ : ℓ P Nu and tφ ℓ : ℓ P Nu be two sequences of function R defined bỹ ϕ ℓ paq "ϕpℓaq, a P R, We also set Now, let α ą d 2 be a fixed number. For each ℓ P N we define a function One can show that since H α Ă L 8 (as α ą d 2 ), the map Ψ ℓ is twice (Fréchet) differentiable 1 and its first and second derivatives satisfy, for d P H α and k, f P H α , and In particular, if d P H α and k, f P H α are such that kpxq K dpxq and fpxq K dpxq for all x P O, 1 One might think that since Φ ℓ is well defined on the space H 1 , it would also be twice differentiable in H 1 . However, for this to hold, we need to restrict it to the space H α for α ą d 2 as in this case H α Ă L 8 . This is fact related to the properties of Nemytski maps, see the papers by the first named authour [5] and [8]. and Note that the stochastic integral vanishes because Gpdps, xqq K dps, xq for all s P r0, τq and x P O.
Using the identities ∇|d| 2 " 2∇dd, (5.14) Observe also that since ∇¨v " 0 we have 2 ż Bearing in mind the two remarks above, we infer that for every m P N, ypt^τ m q satisfies for all t P r0, Ts, P-a.s., Since the second term in the left hand side of the above inequality is positive and yp0q " p|d 0 | 2´1 q´ 2 and by assumption |d 0 px, ωq| 2 " 1 for all x P O and ω P Ω we deduce that, for every m P N, for all t P r0, Ts, P-a.s., ypt^τ m q " 0.
Since H α`1 Ă C 1 pOq as α ą d 2 , we infer that for all m P N, t P r0, Ts, P-a.s. p|dpt^τ m , x, ωq| 2´1 q´" 0 for all x P O. (5.16) Thus, to complete the proof it is sufficient to show that for all m P N, for all t P r0, Ts we have , P-a.s.
p|dpt^τ m , xq| 2´1 q`" 0 for all x P O. (5.17) For this purpose we set ξpt, xq :"`|dpt, xq| 2´1˘`, pt, xq P r0, τqˆO, zptq " ξptq 2 L 2 , t P r0, Ts, and construct a sequence of functions Ψ ℓ very similar to the one defined in (5.6). First let us define an increasing function ϕ : R Ñ r0, 1s belonging to C 8 satisfying Taking the expectation (over the set Ω m,N ), because for a nonnegative function z, ş t^τ 0 zpsq ds ď ş t 0 zps^τq ds, from the above inequality we get Applying the Gronwall Lemma we infer that Hence we infer that 1 Ω m,N p|dpt^τ m q| 2´1 q`" 0 for every t P r0, Ts, P-a.s. and therefore we deduce that for every t P r0, Ts and for every ε ą 0 P´p|dt^τ m | 2´1 q`" 0¯ě 1´ε.
From this last estimate and the first part of the proof infer that for all t P r0, Ts, m P N, P-a.s.
Appendix A. Local strong solution for an abstract stochastic evolution equation.
The goal of this section is to recall general results about the existence of a local and maximal solution to an abstract stochastic partial differential equation with locally Lipschitz continuous coefficients. These results were proved in [7] utilising some truncation and fixed point methods. The proofs are highly technical, and hence we refer the reader to [7] for the details.
To start with let us fix some notations and assumptions. Let V, E and H be separable Banach spaces such that E Ă V continuously. We denote the norm in V by }¨} and we put X T :" Cpr0, Ts; Vq X L 2 p0, T; Eq (A.1) with the norm |¨| X T satisfying |u| 2 X T " sup sPr0,Ts Let F and G be two nonlinear mappings satisfying the following sets of conditions. Assumptions 2. Suppose that F : E Ñ H is such that Fp0q " 0 and there exist p, q ě 1, α, γ P r0, 1q and C ą 0 such that for any x, y P E.
Let K be a separable Hilbert space and L 2 pK, Vq the space of Hilbert-Schmidt operators from K onto V. For the sake of simplicity we denote by ¨ L 2 the norm in L 2 pK, Vq.
Assumptions 3. Assume that G : E Ñ L 2 pK, Vq such that Gp0q " 0 and there exists k ě 1, β P r0, 1q and C G ą 0 such that for any x, y P E.
Let pΩ, F , Pq be a complete probability space equipped with a filtration F " tF t : t ě 0u satisfying the usual condition. By M 2 pX T q we denote the space of all progressively measurable E-values processes whose trajectories belong to X T almost surely endowed with a norm |¨| M 2 pX T q satisfying Let us also formulate the following assumptions.

Assumptions 4.
Suppose that the embeddings E Ă V Ă H are continuous. Consider (for simplicity) a one-dimensional Wiener process W " tWptq : t ě 0u. Assume that tSptq : t P r0, 8qu, is a family of bounded linear operators on the space H such that there exists two positive constants C 1 and C 2 with the following properties: (i) For every T ą 0 and every f P L 2 p0, T; Hq a function u " S˚f defined by uptq " ż T 0 Spt´rq f prq dt, t P r0, Ts, belongs to X T and |u| X T ď C 1 | f | L 2 p0,T;Hq . (A.6) (ii) For every T ą 0 and every process ξ P M 2 p0, T; L 2 pK, Vqq a process u " S˛ξ defined by uptq " Spt´rqξprq dWprq, t P r0, Ts belongs to M 2 pX T q and |u| M 2 pX T q ď C 2 |ξ| M 2 p0,T;L 2 pK,Vqq .
(iii) For every T ą 0 and every u 0 P V, a function u " Su 0 defined by uptq " Sptqu 0 , t P r0, Ts belongs to X T . Moreover, for every T 0 ą 0 there exist C 0 ą 0 such that for all T P p0, T 0 s, |u| X T ď C 0 }u 0 }. (A.7) Now let us consider a semigroup tSptq : t P r0, 8qu as above and the abstract SPDEs  duptq " Auptq dt`F`uptq˘dt`G`uptq˘dWptq, t ą 0, up0q " u 0 . (A.9) Here A is the infinitesimal generator of the semigroup tSptq : t ě 0u. We will not recall the definitions of local and maximal solutions since they are the same as the ones introduced definition 3.3 and definition 3.4. We directly give the main theorems that are of interest to us. The first one is about the existence and uniqueness of a local solution and a probabilistic lower bound of the solution's lifespan.
Theorem A.1. Suppose that Assumption 2, Assumption 3, and Assumption 4 hold. Then for every F 0 -measurable V-valued square integrable random variable u 0 there exists a local process u "`uptq, t P r0, T 1 q˘which is the unique local mild solution to our problem. Moreover, given R ą 0 and ε ą 0 there exists a stopping time τpε, Rq ą 0, such that for every F 0 -measurable V-valued random variable u 0 satisfying E}u 0 } 2 ď R 2 , one has P`T 1 ě τpε, Rq˘ě 1´ε.
The next result is about the existence and uniqueness of a maximal solution and the characterization of its lifespan.
The proofs of both theorems are highly nontrivial and technical, we refer to [7,Section 5] for the details.