## Periodic Reporting for period 1 - SELEs (Stochastic Ericksen-Leslie Equations)

Reporting period: 2019-01-07 to 2021-01-06

Molecules in nematic liquid crystals (NLC) are long and thin and have no positional order (like a fluid), but they tend

to align along a direction called the optical director or director. This director can be easily distorted and aligned to form a specific pattern using an external control with intensity above a certain threshold value. This passage from one stable to another stable state, possibly with higher energy, caused by an external force or control is called the Fréedericksz transition and it plays an important role in many branches of applied sciences and the industry of LCDs.

Although the effect of noise on the dynamics of the optical director has been the subject of numerous theoretical and experimental studies, there are still many questions that remain unsolved. For instance, in previous studies in physics the fluid velocity is assumed to be negligible, hence understanding of the simultaneous effects of the noise and the fluid velocity on the Fréedericksz transition is an open problem which is also challenging. One should notice that De Gennes and Prost pointed out that the fluid velocity plays an important role in the dynamics of the optical director. Recently, several mathematical papers such as the pioneering work of Hong, and Lin, Lin and Wang have taken into account the effects of the fluid velocity, but their equations contain neither deterministic nor stochastic external forces. It is the desire to solve these unchallenged problems that formed the impetus behind the present research project. The main objective of the project was to o give a sound mathematical description

of the noise-induced Fréedericksz transition in Nematic Liquid Crystal (NLC) with general geometric configurations. In particular, we aimed to solve some important and difficult open mathematical problems related to the stochastic partial differential equations (SPDEs) describing the dynamics of noise driven NLC, and give a rigorous mathematical proof of the noise-induced Fréedericksz transition in NLC.

to align along a direction called the optical director or director. This director can be easily distorted and aligned to form a specific pattern using an external control with intensity above a certain threshold value. This passage from one stable to another stable state, possibly with higher energy, caused by an external force or control is called the Fréedericksz transition and it plays an important role in many branches of applied sciences and the industry of LCDs.

Although the effect of noise on the dynamics of the optical director has been the subject of numerous theoretical and experimental studies, there are still many questions that remain unsolved. For instance, in previous studies in physics the fluid velocity is assumed to be negligible, hence understanding of the simultaneous effects of the noise and the fluid velocity on the Fréedericksz transition is an open problem which is also challenging. One should notice that De Gennes and Prost pointed out that the fluid velocity plays an important role in the dynamics of the optical director. Recently, several mathematical papers such as the pioneering work of Hong, and Lin, Lin and Wang have taken into account the effects of the fluid velocity, but their equations contain neither deterministic nor stochastic external forces. It is the desire to solve these unchallenged problems that formed the impetus behind the present research project. The main objective of the project was to o give a sound mathematical description

of the noise-induced Fréedericksz transition in Nematic Liquid Crystal (NLC) with general geometric configurations. In particular, we aimed to solve some important and difficult open mathematical problems related to the stochastic partial differential equations (SPDEs) describing the dynamics of noise driven NLC, and give a rigorous mathematical proof of the noise-induced Fréedericksz transition in NLC.

The project had four main outcomes

1) We developed a general method based on fixed point arguments to establish the existence and uniqueness of a maximal local solution of an abstract stochastic evolution equations with coefficients satisfying local Lipschitz condition involving the norms of two different Banach spaces. We applied this method to prove the existence of a unique maximal local strong solutions to a stochastic system for both 2D and 3D penalised nematic liquid crystals driven by multiplicative Gaussian noise. This joint work of the ER, the supervisor and Prof. Erika Hausenblas has been accepted for publication in the Indiana University Mathematics Journal and also appeared as a preprint. e-Print: arXiv:2004.00590. (see Publication [1])

2) We initiated the mathematical investigation of the Stochastic Ericksen-Leslie equations. In particular, we proved 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. These results were published in Discrete and Continuous Dynamical Systems-B in a joint publication of the ER, the supervisor and Prof Erika Hausenblas. DOI: 10.3934/dcdsb.2019106. (see Publication [2])

3) We firstly built a mathematical system which describes the hydrodynamics of nematic Liquid crystal with external body forces and anisotropic energy modelling the energy of applied external control such as magnetic or electric field. Secondly, under general assumptions on the initial data, the external data and the anisotropic energy, we proved the existence and uniqueness of global weak solutions with finitely many singular times of our mathematical system. Moreover, we showed that if the initial data and the external forces are sufficiently small, then the global weak solution does not have any singular times and is regular as long as the data are regular. This joint work of the ER, the supervisor and Prof. Gabriel Deugoué has appeared as a preprint and submitted for publication in Journal of Evolution Equations. e-Print: arXiv:2005.07659. (See Publication [3]).

4) The analysis of the convergence of solutions to the Ginzburg-Landau approximation of Ericksen-Leslie equations to the original Ericksen-Leslie equations is a very difficult problem and offers several open problem even for the deterministic PDEs. We initiated this analysis for the case of stochastic Ericksen-Leslie equations (SELEs). In particular, we constructed local regular solutions to SELEs by studying the convergence of a sequence of solutions to the Ginzburg-Landau approximation of SELEs with sufficiently smooth initial data and in smooth driving noise. This joint work of the ER, the supervisor and Prof. Gabriel Deugoué has appeared as a preprint and submitted for publication in Journal of Evolution Equations. e-Print: arXiv:2011.00100. ( See Publication [4]).

References:

[1] Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby Strong solution to stochastic penalised nematic liquid crystals model driven by multiplicative Gaussian noise, 41 pages, To appear in The Indiana University Mathematics Journal, preprint available at https://arxiv.org/abs/2004.00590

[2] Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby. A note on the Stochastic Ericksen-Leslie equations for nematic liquid crystals, Discrete and Continuous Dynamical Systems Serie B, 24(11): 5785–5802, 2019, DOI:10.3934/dcdsb.2019106. Preprint available at https://arxiv.org/abs/1902.09245

[3] Z. Brzeźniak, G. Deugoué and P. A. Razafimandimby, On the 2D Ericksen-Leslie equations with anisotropic energy and external forces, 52 pages, preprint available at https://arxiv.org/abs/2005.07659.

[4] Z. Brzeźniak, G. Deugoué and P. A. Razafimandimby, On strong solution to the 2D stochastic Ericksen-Leslie system: A Ginzburg-Landau approximation approach, 16 pages, Preprint available at https://arxiv.org/abs/2011.00100.

1) We developed a general method based on fixed point arguments to establish the existence and uniqueness of a maximal local solution of an abstract stochastic evolution equations with coefficients satisfying local Lipschitz condition involving the norms of two different Banach spaces. We applied this method to prove the existence of a unique maximal local strong solutions to a stochastic system for both 2D and 3D penalised nematic liquid crystals driven by multiplicative Gaussian noise. This joint work of the ER, the supervisor and Prof. Erika Hausenblas has been accepted for publication in the Indiana University Mathematics Journal and also appeared as a preprint. e-Print: arXiv:2004.00590. (see Publication [1])

2) We initiated the mathematical investigation of the Stochastic Ericksen-Leslie equations. In particular, we proved 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. These results were published in Discrete and Continuous Dynamical Systems-B in a joint publication of the ER, the supervisor and Prof Erika Hausenblas. DOI: 10.3934/dcdsb.2019106. (see Publication [2])

3) We firstly built a mathematical system which describes the hydrodynamics of nematic Liquid crystal with external body forces and anisotropic energy modelling the energy of applied external control such as magnetic or electric field. Secondly, under general assumptions on the initial data, the external data and the anisotropic energy, we proved the existence and uniqueness of global weak solutions with finitely many singular times of our mathematical system. Moreover, we showed that if the initial data and the external forces are sufficiently small, then the global weak solution does not have any singular times and is regular as long as the data are regular. This joint work of the ER, the supervisor and Prof. Gabriel Deugoué has appeared as a preprint and submitted for publication in Journal of Evolution Equations. e-Print: arXiv:2005.07659. (See Publication [3]).

4) The analysis of the convergence of solutions to the Ginzburg-Landau approximation of Ericksen-Leslie equations to the original Ericksen-Leslie equations is a very difficult problem and offers several open problem even for the deterministic PDEs. We initiated this analysis for the case of stochastic Ericksen-Leslie equations (SELEs). In particular, we constructed local regular solutions to SELEs by studying the convergence of a sequence of solutions to the Ginzburg-Landau approximation of SELEs with sufficiently smooth initial data and in smooth driving noise. This joint work of the ER, the supervisor and Prof. Gabriel Deugoué has appeared as a preprint and submitted for publication in Journal of Evolution Equations. e-Print: arXiv:2011.00100. ( See Publication [4]).

References:

[1] Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby Strong solution to stochastic penalised nematic liquid crystals model driven by multiplicative Gaussian noise, 41 pages, To appear in The Indiana University Mathematics Journal, preprint available at https://arxiv.org/abs/2004.00590

[2] Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby. A note on the Stochastic Ericksen-Leslie equations for nematic liquid crystals, Discrete and Continuous Dynamical Systems Serie B, 24(11): 5785–5802, 2019, DOI:10.3934/dcdsb.2019106. Preprint available at https://arxiv.org/abs/1902.09245

[3] Z. Brzeźniak, G. Deugoué and P. A. Razafimandimby, On the 2D Ericksen-Leslie equations with anisotropic energy and external forces, 52 pages, preprint available at https://arxiv.org/abs/2005.07659.

[4] Z. Brzeźniak, G. Deugoué and P. A. Razafimandimby, On strong solution to the 2D stochastic Ericksen-Leslie system: A Ginzburg-Landau approximation approach, 16 pages, Preprint available at https://arxiv.org/abs/2011.00100.

All the completed projects described in the previous section of the report represent progress beyond the state of the art.

* this project is the first mathematical study which presents a systematic mathematical study of the Ericksen-Leslie equations with anisotropic energy and body forces. This is important as

*As mentioned in the previous section the current project is the first to a sound mathematical analysis of the convergence of the stochastic Ginzburg-Landau approximation of the Stochastic Ericksen-Leslie equations.

* When completed, our current project will be the foundation of a rigorous mathematical formulation and proof of the noise-induced Fréedericksz transition in NLC.

This project is of theoretical nature and without direct technological applications in view. However, the Fréedericksz transition is exploited in the industry of Liquid crystal Displays (LCDs), there might be a potential application of our theoretical in the manufacture of LCDs.

During the first year of the fellowship, i.e. 2019, the ER delivered over 6 talks in invited seminars as well as conferences. For example, he was a Plenary speaker at the Workshop “Recent trends in Stochastic Analysis and Partial Differential Equations”, University of Chester, an invited speaker at the LMS-East Midlands Stochastic Analysis Seminar, University of York, and delivered seminar talks at the University of Leoben, University of Pretoria.

* this project is the first mathematical study which presents a systematic mathematical study of the Ericksen-Leslie equations with anisotropic energy and body forces. This is important as

*As mentioned in the previous section the current project is the first to a sound mathematical analysis of the convergence of the stochastic Ginzburg-Landau approximation of the Stochastic Ericksen-Leslie equations.

* When completed, our current project will be the foundation of a rigorous mathematical formulation and proof of the noise-induced Fréedericksz transition in NLC.

This project is of theoretical nature and without direct technological applications in view. However, the Fréedericksz transition is exploited in the industry of Liquid crystal Displays (LCDs), there might be a potential application of our theoretical in the manufacture of LCDs.

During the first year of the fellowship, i.e. 2019, the ER delivered over 6 talks in invited seminars as well as conferences. For example, he was a Plenary speaker at the Workshop “Recent trends in Stochastic Analysis and Partial Differential Equations”, University of Chester, an invited speaker at the LMS-East Midlands Stochastic Analysis Seminar, University of York, and delivered seminar talks at the University of Leoben, University of Pretoria.