Mathematical Analysis of the Stochastic Nematic Liquid Cristal

The effect of random external perturbation on the dynamics of the nematic liquid crystals has been the subject of numerous theoretical and experimental studies in Chemistry, Engineering and Physics. Many prominent scientists have shown, for instance, that in the presence of noise the liquid crystal will leave an unstable state quite fast and goes to a stable one. However, these previous works just looked at the alignment of the molecules and have neglected the effect of the hydrodynamic flow. However, it is pointed out by de Gennes and J.~Prost in the book The Physics of Liquid Crystals, published by Clarendon Press, Oxford in 1993, that the fluid flow disturbs the alignment and conversely a change in the alignment will induce a flow in the nematic liquid crystal. Hence for a full understanding of the effect of fluctuating external fields on the behavior of the liquid crystals one needs to take into account the dynamics of director field and the velocity of the fluid. Motivated by these facts, in this project we intend to initiate and perform mathematical analysis on stochastic equations arisen from the hydrodynamics of incompressible nematic liquid crystals under the influence of random external perturbations. Our first aim is to study basic questions such as the existence and uniqueness of solution of the above system and related problems. The second aim will be the investigation of the long-time behavior of the above system. Here we are mainly interested in the existence of stationary solution or invariant measure. We will also study the uniqueness of the invariant measure. The last but not least aim is to analyze and implement some numerical approximations of our stochastic models.

Nematic Liquid crystal (LC) is a state of matter that has properties between those of con- ventional liquid and those of solid crystal. It may flow like a liquid, but its molecules may be oriented in a crystal-like way. The direction along which these molecules are oriented is called the optical director and is usually denoted by a unit vector n. The effect of external random perturbations on the dynamics of the optical director of nematic liquid crystals has been the subject of numerous theoretical and experimental studies in Chemistry, Engineering and Physics. Many prominent scientists have shown that, for instance, the decay time needed by the system to leave an unstable state diminishes in presence of a fluctuating magnetic field. However, these previous works have merely taken into account the alignment due to the optical director and have neglected the effect of the hydrodynamic flow. In particular, it is pointed out in [P. G. de Gennes and J. Prost, The Physics of Liquid Crystals. Clarendon Press, Oxford 1993.] that the fluid flow disturbs the alignment of the optical director field and conversely, a change of the alignment of the optical director field will induce a flow in the nematic liquid crystal. Hence, it is desirable to have a theory or a model that takes into account the dynamics of n and the velocity field v of the fluid when investigating the effect of fluctuating external fields on the behaviour of the liquid crystals. The project was motivated by these facts mentioned above. The aim is to analyse the stochastic models of the hydrodynamics of incompressible nematic liquid crystals under the influence of random external perturbations.