No Access

On the uniqueness of weak solutions to the Ericksen–Leslie liquid crystal model in $ℝ^{2}$

Jinkai Li

Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

E-mail Address: jklimath@gmail.com

Search for more papers by this author

Edriss S. Titi

Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843-3368, USA

E-mail Address: titi@math.tamu.edu

E-mail Address: edriss.titi@weizmann.ac.il

Search for more papers by this author

, and

Zhouping Xin

The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong, P. R. China

E-mail Address: zpxin@ims.cuhk.edu.hk

Search for more papers by this author

https://doi.org/10.1142/S0218202516500184Cited by:32 (Source: Crossref)

Abstract

This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen–Leslie system of liquid crystal models in $ℝ^{2}$ , with both general Leslie stress tensors and general Oseen–Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant. One of the main ideas of our proof is to perform suitable energy estimates at the level one order lower than the natural basic energy estimates for the Ericksen–Leslie system.

Communicated by N. Bellomo

Keywords:

AMSC: 76D03, 35D30, 76A15

References

1. P. Constantin and W. Sun, Remarks on Oldroyd-B and related complex fluid models, Commun. Math. Sci. 10 (2012) 33–73. Crossref, Web of Science, Google Scholar
2. J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol. 5 (1961) 23–34. Crossref, Google Scholar
3. M.-C. Hong, Global existence of solutions of the simplified Ericksen–Leslie system in dimension two, Calc. Var. Partial Differential Equations 40 (2011) 15–36. Crossref, Web of Science, Google Scholar
4. M.-C. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in $ℝ^{2}$ , Adv. Math. 231 (2012) 1364–1400. Crossref, Web of Science, Google Scholar
5. J. Huang, F.-H. Lin and C. Wang, Regularity and existence of global solutions to the Ericksen–Leslie system in $ℝ^{2}$ , Commun. Math. Phys. 331 (2014) 805–850. Crossref, Web of Science, Google Scholar
6. A. Larios, E. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations 255 (2013) 2636–2654. Crossref, Web of Science, Google Scholar
7. F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal. 28 (1968) 265–283. Crossref, Web of Science, Google Scholar
8. F.-H. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal. 197 (2010) 297–336. Crossref, Web of Science, Google Scholar
9. F.-H. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Ann. Math. Ser. B 31 (2010) 921–938. Crossref, Web of Science, Google Scholar
10. M. Wang and W. Wang, Global existence of weak solution for the 2D Ericksen–Leslie system, Calc. Var. Partial Differential Equations 51 (2014) 915–962. Crossref, Web of Science, Google Scholar
11. M. Wang, W. Wang and Z. Zhang, On the uniqueness of weak solutions for the 2D Ericksen–Leslie system, preprint. Google Scholar