On the Weak Solutions to a Stochastic 2D Simplified Ericksen-Leslie Model

Abstract

We study in this article a stochastic version of a 2D Ericksen-Leslie systems. The system model the dynamic of nematic liquid crystals under the influence of stochastic external forces and stretching effects. We prove the existence of a probabilistic weak solutions. The proof relies on a reformulation of the model proposed in Gong et al. (Nonlinearity 28(10), 3677–3694 2015) as well as a Galerkin approximation and some compactness results. We also prove the pathwise uniqueness of the weak solution when the stretching effect is neglected.

Appendix

In this section, we recall from [1, 38, 41] some important compactness results used in this article. Let E be a Banach space and let $\mathcal {B}(E)$ be its Borel σ −field

The following lemma is borrowed from [1, 41].

Lemma A.1

LetX, YandZbe some Banach spacessuch thatXis compactlyembedded intoYand letYbea subset ofZ. For any 1 ≤ p, q ≤∞, letMbe a setbounded inL^q(0, T; X) suchthat

$$\displaystyle \lim\limits_{\delta \rightarrow 0} {\displaystyle {\int}^{T}_{0}} \Arrowvert v(t + \delta) - v(t) {\Arrowvert_{Z}^{p}} dt = 0, \ \text{ uniformly for all } v \in M. $$

Then Mis relatively compact in L^p(0, T; Y ).

We recall the following concept of tightness of probability measures.

Definition A.1

A family of probability measure $\mathcal {P}$ on $(E, \mathcal {B}(E))$ is tight if for arbitrary 𝜖 > 0, there exists a compact set K_𝜖 ⊂ E such that

$$\mu(K_{\epsilon}) \ge 1 - \epsilon, \ \forall \mu \in \mathcal{P}. $$

A sequence of measure {μ_n} on $(E, \mathcal {B}(E))$ is weakly convergent to a measure μ if for all continuous and bounded functions φ on E

$$\displaystyle \lim_{n \rightarrow \infty } \displaystyle {\int}_{E} \varphi (x) \mu_{n} (dx) = \displaystyle {\int}_{E} \varphi (x) \mu (dx). $$

We recall the following important lemmas, due to Prohorov and Skorokhod.

The following result, which is proven in [39] shows that the tightness property is a compactness criteria.

Lemma A.2

A sequence of measure {μ_n} on$(E, \mathcal {B}(E))$istight if and only if it is relatively compact, that is there exists asubsequence$\{\mu _{n_{k}} \}$which weakly converges to a probability measureμ.

The next result is due to Skorokhod, [42]. It relates the weak convergence of probabilitymeasures with that of almost everywhere convergence of random variables.

Lemma A.3

For an arbitrary sequence of probability {μ_n} on$(E, \mathcal {B}(E))$weakly convergent toa measureμ, there exista probability space$({\Omega }, \bar {\mathfrak {F}}, \mathbb {P})$andrandom variablesζ, ζ₁, ζ₂,⋯ζ_n,⋯ withvalues inEsuch thatthe probability law of$\zeta _{n} \mathcal {L}(\zeta _{n})(A) = \mathbb {P} \{ \omega \in {\Omega }: \zeta _{n}(\omega ) \in A \}, \ \forall A \in \bar {\mathfrak {F}},$isμ_n, the probabilitylaw ofζisμ, and

$$\displaystyle \lim_{n \rightarrow \infty } \zeta_{n} = \zeta, \ \mathbb{P}-a.s. $$