Random attractor for the 2D stochastic nematic liquid crystals flows

Boling Guo; Yongqian Han; Guoli Zhou

doi:10.3934/cpaa.2019106

Article Contents

2019, Volume 18, Issue 5: 2349-2376. Doi: 10.3934/cpaa.2019106

This issue Previous Article Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model Next Article Molecular characterization of anisotropic weak Musielak-Orlicz Hardy spaces and their applications

Random attractor for the 2D stochastic nematic liquid crystals flows

1.
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
2.
No. 174 Sha Zheng street, Shapingba District, Chongqing University, Chongqing 401331, China

^* Corresponding author
^* Corresponding author

Received: February 28, 2018

Revised: November 30, 2018

Published: March 2019

This work was partially supported by NNSF of China(Grant No. 11401057), Natural Science Foundation Project of CQ (Grant No. cstc2016jcyjA0326), Fundamental Research Funds for the Central Universities(Grant No.2018CDXYST0024, 106112015CDJXY100005) and China Scholarship Council (Grant No.:201506055003)

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Abstract

We consider the long-time behavior for stochastic 2D nematic liquid crystals flows with the velocity field perturbed by an additive noise. The presence of the noises destroys the basic balance law of the nematic liquid crystals flows, so we can not follow the standard argument to obtain uniform a priori estimates for the stochastic flow even in the weak solution space under non-periodic boundary conditions. To overcome the difficulty we use a new technique some kind of logarithmic energy estimates to obtain the uniform estimates which improve the previous result for the orientation field that grows exponentially w.r.t.time t. Considering the existence of random attractor, the common method is to derive uniform a priori estimates in functional space which is more regular than the solution space. We can follow the common method to prove the existence of random attractor in the weak solution space. However, if we consider the existence of random attractor in the strong solution space, it is very difficult and very complicated for such highly non-linear stochastic system with no basic balance law and non-periodic boundary conditions. Here, we use a compactness arguments of the stochastic flow and regularity of the solutions to the stochastic model to obtain the existence of the random attractor in the strong solution space, which implies the support of the invariant measure lies in a more regular space. As far as we know, it is the first article to attack the long-time behavior of stochastic nematic liquid crystals.

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Mathematics Subject Classification: Primary: 60H15; Secondary: 35Q35.

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