# American Institute of Mathematical Sciences

January  2014, 13(1): 225-236. doi: 10.3934/cpaa.2014.13.225

## The existence and blow-up criterion of liquid crystals system in critical Besov space

 1 Institute of Mathematics, Hangzhou Dianzi University, Zhejiang, Hangzhou, 310018, China 2 Institute of Mathematics, Fudan University, Shanghai

Received  December 2012 Revised  April 2013 Published  July 2013

We consider the existence of strong solution to liquid crystals system in critical Besov space, and give a criterion which is similar to Serrin's criterion on regularity of weak solution to Navier-Stokes equations.
Citation: Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure and Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225
##### References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations," Springer, 2011. [2] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078. [3] R. Danchin, "Fourier Analysis Methods for PDE's," 2005. Available from: http://www.fichier-pdf.fr/2011/12/13/courschine/courschine.pdf. [4] J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358. [5] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188. [6] M. C. Hong, Global existence of solutions of the simplified Ericksen–Leslie system in dimension two, Calc. Var., 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5. [7] H. Kozono and Y. Shimada, Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations, Math. Nachr., 276 (2004), 63-74. doi: 10.1002/mana.200310213. [8] F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1986), 265-283. doi: 10.1007/BF00251810. [9] X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767. doi: 10.1016/j.jde.2011.08.045. [10] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 1989, 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. [11] F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst. A, 2 (1998), 1-22. [12] F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., XLVIII, 1995, 501-537. doi: 10.1002/cpa.3160480503. [13] F. H. Lin and C. Liu, Existence of Solutions for the Ericksen-Leslie System, Arch. Rational Mech. Anal., 154 (2000), 135-156. doi: 10.1007/s002050000102. [14] F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid Crystal Flows in Two Dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. [15] J. Y. Lin and S. J. Ding, On the well-posedness for the heat flow of harmonic maps and the hydrodynamic flow of nematic liquid crystals in critical spaces, Math. Meth. Appl. Sci., 35 (2012), 158-173. doi: 10.1002/mma.1548. [16] C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Rational Mech. Anal., 200 (2011), 1-19. doi: 10.1007/s00205-010-0343-5. [17] H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345. doi: 10.1007/s00526-011-0460-5.

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##### References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations," Springer, 2011. [2] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078. [3] R. Danchin, "Fourier Analysis Methods for PDE's," 2005. Available from: http://www.fichier-pdf.fr/2011/12/13/courschine/courschine.pdf. [4] J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358. [5] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188. [6] M. C. Hong, Global existence of solutions of the simplified Ericksen–Leslie system in dimension two, Calc. Var., 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5. [7] H. Kozono and Y. Shimada, Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations, Math. Nachr., 276 (2004), 63-74. doi: 10.1002/mana.200310213. [8] F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1986), 265-283. doi: 10.1007/BF00251810. [9] X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767. doi: 10.1016/j.jde.2011.08.045. [10] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 1989, 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. [11] F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst. A, 2 (1998), 1-22. [12] F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., XLVIII, 1995, 501-537. doi: 10.1002/cpa.3160480503. [13] F. H. Lin and C. Liu, Existence of Solutions for the Ericksen-Leslie System, Arch. Rational Mech. Anal., 154 (2000), 135-156. doi: 10.1007/s002050000102. [14] F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid Crystal Flows in Two Dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. [15] J. Y. Lin and S. J. Ding, On the well-posedness for the heat flow of harmonic maps and the hydrodynamic flow of nematic liquid crystals in critical spaces, Math. Meth. Appl. Sci., 35 (2012), 158-173. doi: 10.1002/mma.1548. [16] C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Rational Mech. Anal., 200 (2011), 1-19. doi: 10.1007/s00205-010-0343-5. [17] H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345. doi: 10.1007/s00526-011-0460-5.
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