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

Abstract

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.2010 Mathematics Subject Classification. Primary: 60H15, 37L40; Secondary: 35R60.

On a novel approach for modeling liquid crystalline flows

“…[13,19,6]). For stochastic Ericksen-Leslie equations, we refer to [2] and the references therein. A more extensive survey can be found in [10].…”

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

“…However, it is pointed out in [23,Chapter 5] that the fluid flow disturbs the alignment and conversely a change in the alignment will induce a flow in the nematic liquid crystal. It is this gap in knowledge that is the motivation for our mathematical study which was initiated in the old unpublished preprints [7] and [8], see also the recent papers [6] and [5].…”

“…We should notice that some of the arguments elaborated in Section 5 have been already used in [1] and [5] which respectively studied the strong solution of some stochastic hydrodynamic equations (NSEs, MHD and 3D Leray α-models) driven by Lévy noise, and the existence and uniqueness of a maximal local smooth solution to the stochastic Ericksen-Leslie system (1.1)-(1.4) on the d-dimensional torus. We are also strongly convinced that with these general results it is possible, although it has not been done in detail, to prove the existence of strong solution of several stochastic hydrodynamical models such as the NSEs, MHD equations, α-models for Navier-Stokes and related problems.…”

“…Note that in [26] the second unknown n is a scalar field, the nonlinear term f (·) is globally Lipschitz and the derivative of a double-well potential F (·), and the coefficient of the noise entering the equations for n is bounded. The paper [5] is the first paper to deal with the the stochastic counterpart of the Ericksen-Leslie equations (1.1)- (1.4). The results in present manuscript is not covered in [5] because in contrast to our framework which considers initial condition (v 0 , n 0 ) ∈ H 1 × H 2 , the initial data in [5] satisfies (v 0 , n 0 ) ∈ H α × H α+1 for α > d 2 , where d = 2, 3 is the space dimension.…”

“…The paper [5] is the first paper to deal with the the stochastic counterpart of the Ericksen-Leslie equations (1.1)- (1.4). The results in present manuscript is not covered in [5] because in contrast to our framework which considers initial condition (v 0 , n 0 ) ∈ H 1 × H 2 , the initial data in [5] satisfies (v 0 , n 0 ) ∈ H α × H α+1 for α > d 2 , where d = 2, 3 is the space dimension. There is also the papers [59] which seeks for a special solution (v, n) with the unknown n is replaced by an angle θ such that n = (cos θ, sin θ).…”