GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D

doi:10.1016/S0252-9602(16)30054-6

Acta mathematica scientia,Series B ›› 2016, Vol. 36 ›› Issue (4): 973-1014.doi: 10.1016/S0252-9602(16)30054-6

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GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D

Jinkai LI^1,2, Zhouping XIN¹

1 The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong, China;
2 Department of Computer Science and Applied Mathematics, Weizmann Institute of Science Rehovot 76100, Israel

Received:2016-02-26 Online:2016-08-25 Published:2016-08-25
Contact: Zhouping XIN,E-mail:zpxin@ims.cuhk.edu.hk E-mail:zpxin@ims.cuhk.edu.hk
Supported by:
This research is supported partially by Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants 14305315, CUHK4041/11P and CUHK4048/13P, a Focus Area Grant from The Chinese University of Hong Kong, a Croucher Foundation-CAS Joint Grant, and a NSFC/RGC Joint Research Scheme (N-CUHK443/14).

Abstract

Abstract:

In this article, we prove the global existence of weak solutions to the non-isothermal nematic liquid crystal system on T², on the basis of a new approximate system which is different from the classical Ginzburg-Landau approximation. Local in space energy inequalities are employed to recover the estimates on the second order spatial derivatives of the director fields locally in time, which cannot be derived from the basic energy balance. It is shown that these weak solutions satisfy the temperature equation, and also the total energy equation but away from at most finite many "singular" times, at which the energy concentration occurs and the director field losses its second order derivatives.

Key words: Global weak solutions, non-isothermal, nematic liquid crystals

CLC Number:

35D30

Jinkai LI, Zhouping XIN. GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D[J].Acta mathematica scientia,Series B, 2016, 36(4): 973-1014.

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GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D

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