On the existence and uniqueness of solution to a stochastic simplified liquid crystal model

,T Tachim Medjo

doi:10.3934/cpaa.2019101

Abstract

We study in this article a stochastic version of a 2D simplified Ericksen-Leslie systems, which model the dynamic of nematic liquid crystals under the influence of stochastic external forces. We prove the existence and uniqueness of strong solution. The proof relies on a new formulation of the model proposed in [ 19 ] as well as a Galerkin approximation

Highlights

Stochastic partial differential equations (SPDE) are used to model physical systems subjected to influence of internal, external or environmental noises or to describe systems that are too complex to be described deterministically, e.g. a flow of a chemical substance in a river subjected by wind and rain, an airflow around an airplane wing perturbed by the random state of the atmosphere and weather, a laser beam subjected to turbulent movement of the atmosphere, spread of an epidemic in some regions and the spatial spread of infectious diseases
The 2-dimensional Navier-Stokes equations with sufficiently degenerate noise have a unique invariant measure and exhibit ergodic behavior in the sense that the time average of a solution is equal to the average over all possible initial data
As recalled in [20], the mathematical studies on the dynamical liquid crystal systems started with the work of [33, 35, 32, 34], where the authors established the global existence of weak solutions, in both 2D and 3D, to the Ginzburg-Landau approximation of the liquid crystal system, see [8, 50] for some generalizations to the general liquid crystal systems

Summary

Introduction

Stochastic partial differential equations (SPDE) are used to model physical systems subjected to influence of internal, external or environmental noises or to describe systems that are too complex to be described deterministically, e.g. a flow of a chemical substance in a river subjected by wind and rain, an airflow around an airplane wing perturbed by the random state of the atmosphere and weather, a laser beam subjected to turbulent movement of the atmosphere, spread of an epidemic in some regions and the spatial spread of infectious diseases. As recalled in [20], the mathematical studies on the dynamical liquid crystal systems started with the work of [33, 35, 32, 34], where the authors established the global existence of weak solutions, in both 2D and 3D, to the Ginzburg-Landau approximation of the liquid crystal system, see [8, 50] for some generalizations to the general liquid crystal systems. In [42], the author has studied the stationary orientational correlations of the director field of a nematic liquid crystal near the Freedericksz transition In this transition the molecules tend to reorient due to some random external perturbations. In [4], the authors proved several results (including the existence and uniqueness of weak solutions) of a simplified stochastic EL model with Ginzburg-Landau approximation. The positive constants ν1, λ, ν2 are the viscosity of the fluid, the competition between the kinetic and the potential energy amd the microscopic elastic relaxation time, respectively

The symbol

Now we define the Hilbert spaces H and U by

We also note that

It follows that t

We also have τn lim E