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

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

数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (4): 973-1014.doi: 10.1016/S0252-9602(16)30054-6

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

李进开^1,2, 辛周平¹

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

收稿日期:2016-02-26 出版日期:2016-08-25 发布日期:2016-08-25
通讯作者: Zhouping XIN,E-mail:zpxin@ims.cuhk.edu.hk E-mail:zpxin@ims.cuhk.edu.hk
作者简介:Jinkai LI,E-mail:jklimath@gmail.com
基金资助:
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).

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).

摘要/Abstract

摘要：

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.

关键词: Global weak solutions, non-isothermal, nematic liquid crystals

Abstract:

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

中图分类号:

35D30

李进开, 辛周平. GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D[J]. 数学物理学报(英文版), 2016, 36(4): 973-1014.

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.

参考文献

[1] Ericksen J. Conservation laws for liquid crystals. Trans Soc Rheol, 1961, 5:22-34
[2] Leslie F M. Some constitutive equations for liquid crystals. Arch Rational Mech Anal, 1968, 28:265-283
[3] Leslie F M. Thermal effects in cholesteric liquid crystals. Proc Roy Soc London A, 1968, 307:359-372
[4] Oswald, P, Pieranski P. Nematic and Cholesteric Liquid Crystals:Concepts and Physical Properties Illustrated by Experiments. CRC Press, 2005
[5] Sonnet A M, Virga E G. Dissipative Ordered Fluids:Theories for Liquid Crystals. New York, Dordrecht, Heidelberg London:Springer, 2012
[6] Feireisl E, Fremond M, Rocca E, Shimperna G. A new approch to non-isothermal models for nematic liquid crystals. Arch Rational Mech Anal, 2012, 205:651-672
[7] Lin F-H. Nonliner theory of defets in nematic liquid crystals:Phase transition and flow phenomena. Comm Pure Appl Math, 1989, 42:789-814
[8] Lin F-H, Liu C. Nonparabolic dissipative systems modeling the flow of liquid crystals. Comm Pure Appl Math, 1995, 48:501-537
[9] Cavaterra C, Rocca E, Wu H. Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows. J Differential Equations, 2013, 255:24-57
[10] Feireisl E, Rocca E, Schimperna G. On a non-isothermal model for nematic liquid crystals. Nonlinearity, 2011, 24:243-257
[11] Wu H, Xu X, Liu C. On the general Ericksen-Leslie system:Parodi's relation, well-posedness and stability. Arch Ration Mech Anal, 2013, 208:59-107
[12] Lin F-H, Liu C. Partial regularity of the dynamic system modeling the flow of liquid crystals. Disc Cont Dyn Sys, 1996, 2:1-22
[13] Jiang F, Tan Z. Global weak solution to the flow of liquid crystals system. Math Meth Appl Sci, 2009, 32:2243-2266
[14] Liu X, Zhang Z. Existence of the flow of liquid crystal system. Chin Ann Math, 2009, 30A:1-20
[15] Chu Y, Hao Y, Liu X. Global weak solutions to a general liquid crystals system. Discrete Contin Dyn Syst Ser A, 2013, 33:2681-2710
[16] Wang D, Yu C. Global weak solution and large time behavior for the compressible flow of liquid crystals. Arch Rational Mech Anal, 2012, 204:881-915
[17] Lin F-H, Liu C, Wang C. Liquid crystal flow in two dimensions. Arch Rational Mech Anal, 2010, 197:297-336
[18] Hong M-C. Global existence of solutions of the simplified Ericksen-Leslie system in dimension two. Calc Var Partial Differential Equations, 2011, 40:15-36
[19] Hong M-C, Xin Z. Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in R². Adv Math, 2012, 231:1364-1400
[20] Wang W, Wang M. Global existence of weak solution for the 2-D Ericksen-Leslie system. Calc Var Partial Differential Equations, 2014, 51:915-962
[21] Huang J, Lin F-H, Wang C. Regularity and existence of global solutions to the Ericksen-Leslie system in R². Comm Math Physics, 2014, 331:805-850
[22] Struwe M. The existence of surfaces of constant mean curvature with free boundaries. Acta Math, 1988, 160:19-64
[23] Lin F, Wang C. On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. Chin Ann Math, 2010, 31B:921-938
[24] Wang M, Wang W, Zhang Z. On the uniqueness of weak solutions for the 2-D Ericksen-Leslie system. Discrete Contin Dyn Syst Ser B, 2016, 21:919-941
[25] Li J, Titi E S, Xin Z. On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in R². Math Models Methods Appl Sci, 2016, 26:803-822
[26] Struwe M. On the evolution of harmonic mappings of Riemannian surfaces. Comment Math Helvetici, 1985, 60:558-581
[27] Simon J. Compact sets in the space L^p(0, T; B). Ann Mat Pure Appl, 1987, 146:65-96
[28] Wang L. On Korn's inequality. J Comput Math, 2003, 21:321-324
[29] Damascelli L. Comparison theorems for some quasilinear degenerate eliptic operators and applications to symmetry and monotonicity results. Ann Inst Poincaré, 1998, 15:493-516
[30] Adams R A, Fournier J F. Sobolev spaces. Second Edition. Pure and Applied Mathematics (Amsterdam), 140. Amsterdam:Elsevier/Academic Press, 2003
[31] Schoen R, Uhlenbeck K. Boundary regularity and the Dirichlet Problem for harmonic maps. J Differential Geom, 1983, 18:253-268
[32] Hong M-C, Li J, Xin Z. Blow-up criteria of strong solutions to the Ericksen-Leslie system in R³. Comm Partial Differential Equations, 2014, 39:1284-1328

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

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

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