Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 37-55, 2017

The Cauchy problem for the liquid crystals system in the critical Besov space with negative index

Sen Ming, Han Yang, Zili Chen, Ls Yong

Received May 13, 2015.   First published February 24, 2017.

Sen Ming, Han Yang, Zili Chen, Department of Mathematics, Southwest Jiaotong University, Two Ring Road, No. 111, Chengdu, 610031, Sichuan, China, e-mail: hanyang95@263.net; Ls Yong, Department of Mathematics, Southwestern University of Finance and Economics, Guanghua Village Street, No. 55, Chengdu, 611130, Sichuan, China

Abstract: The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space $\dot{B}_{p,1}^{n/p-1}(\mathbb R^n)\times\dot{B}_{p,1}^{n/p}(\mathbb R^n)$ with $n<p<2n$ is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed.

Keywords: liquid crystals system; critical Besov space; negative index; well-posedness; blow-up

Classification (MSC 2010): 35Q35, 76A15, 35B44

DOI: 10.21136/CMJ.2017.0249-15

Full text available as PDF.


References:
  [1] H. Abidi: Equation de Navier-Stokes avec densit√© et viscosit√© variables dans l'espace critique. Rev. Mat. Iberoam. 23 (2007), 537-586 French. DOI 10.4171/RMI/505 | MR 2371437 | Zbl 1175.35099
  [2] H. Abidi, G. Gui, P. Zhang: On the wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces. Arch. Ration. Mech. Anal. 204 (2012), 189-230. DOI 10.1007/s00205-011-0473-4 | MR 2898739 | Zbl 1314.76021
  [3] H. Abidi, P. Zhang: On the global well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity. Available at Arxiv:1301.2371.
  [4] H. Bahouri, J.-Y. Chemin, R. Danchin: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften 343, Springer, Heidelberg (2011). DOI 10.1007/978-3-642-16830-7 | MR 2768550 | Zbl 1227.35004
  [5] M. Cannone: Harmonic analysis tools for solving the incompressible Navier-Stokes equations. Handbook of Mathematical Fluid Dynamics. Vol. III Elsevier/North Holland, Amsterdam S. Friedlander et al. (2004), 161-244. DOI 10.1016/s1874-5792(05)80006-0 | MR 2099035 | Zbl 1222.35139
  [6] C. Cavaterra, E. Rocca, H. Wu: Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows. J. Differ. Equations 255 (2013), 24-57. DOI 10.1016/j.jde.2013.03.009 | MR 3045633 | Zbl 1282.35087
  [7] Q. Chen, C. Miao: Global well-posedness for the micropolar fluid system in critical Besov spaces. J. Differ. Equations 252 (2012), 2698-2724. DOI 10.1016/j.jde.2011.09.035 | MR 2860636 | Zbl 1234.35193
  [8] R. Danchin: Local theory in critical spaces for compressible viscous and heat-conductive gases. Commun. Partial Differ. Equations 26 (2001), 1183-1233. DOI 10.1081/PDE-100106132 | MR 1855277 | Zbl 1007.35071
  [9] R. Danchin, P. B. Mucha: A Lagrangian approach for the incompressible Navier-Stokes equations with variable density. Commun. Pure Appl. Math. 65 (2012), 1458-1480. DOI 10.1002/cpa.21409 | MR 2957705 | Zbl 1247.35088
  [10] Y. Du, K. Wang: Regularity of the solutions to the liquid crystal equations with small rough data. J. Differ. Equations 256 (2014), 65-81. DOI 10.1016/j.jde.2013.07.066 | MR 3115835 | Zbl 1320.35125
  [11] J. L. Ericksen: Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal. 9 (1962), 371-378. DOI 10.1007/bf00253358 | MR 0137403 | Zbl 0105.23403
  [12] H. Fujita, T. Kato: On the Navier-Stokes initial value problem. I. Arch. Ration. Mech. Anal. 16 (1964), 269-315. DOI 10.1007/BF00276188 | MR 0166499 | Zbl 0126.42301
  [13] Y. Hao, X. Liu: The existence and blow-up criterion of liquid crystals system in critical Besov space. Commun. Pure Appl. Anal. 13 (2014), 225-236. DOI 10.3934/cpaa.2014.13.225 | MR 3082558 | Zbl 1273.76352
  [14] M.-C. Hong: Global existence of solutions of the simplified Ericksen-Leslie system in dimension two. Calc. Var. Partial Differ. Equ. 40 (2011), 15-36. DOI 10.1007/s00526-010-0331-5 | MR 2745194 | Zbl 1213.35014
  [15] J. Huang, M. Paicu, P. Zhang: Global solutions to 2-D inhomogeneous Navier-Stokes system with general velocity. J. Math. Pures Appl. (9) 100 (2013), 806-831. DOI 10.1016/j.matpur.2013.03.003 | MR 3125269 | Zbl 1290.35184
  [16] F. Jiang, S. Jiang, D. Wang: Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions. Arch. Ration. Mech. Anal. 214 (2014), 403-451. DOI 10.1007/s00205-014-0768-3 | MR 3255696 | Zbl 1307.35225
  [17] X. Li, D. Wang: Global solution to the incompressible flow of liquid crystals. J. Differ. Equations 252 (2012), 745-767. DOI 10.1016/j.jde.2011.08.045 | MR 2852225 | Zbl 1277.35121
  [18] F. Lin: Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun. Pure Appl. Anal. 42 (1989), 789-814. DOI 10.1002/cpa.3160420605 | MR 1003435 | Zbl 0703.35173
  [19] F. Lin, J. Lin, C. Wang: Liquid crystal flows in two dimensions. Arch. Ration. Mech. Anal. 197 (2010), 297-336. DOI 10.1007/s00205-009-0278-x | MR 2646822 | Zbl 1346.76011
  [20] F. Lin, C. Liu: Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete Contin. Dyn. Syst. 2 (1996), 1-22. DOI 10.3934/dcds.1996.2.1 | MR 1367385 | Zbl 0948.35098
  [21] F. Lin, C. Liu: Existence of solutions for the Ericksen-Leslie system. Arch. Ration. Mech. Anal. 154 (2000), 135-156. DOI 10.1007/s002050000102 | MR 1784963 | Zbl 0963.35158
  [22] J. Lin, S. Ding: On the well-posedness for the heat flow of harmonic maps and the hydrodynamic flow of nematic liquid crystals in critical spaces. Math. Methods Appl. Sci. 35 (2012), 158-173. DOI 10.1002/mma.1548 | MR 2876822 | Zbl 1242.35006
  [23] Q. Liu, T. Zhang, J. Zhao: Global solutions to the 3D incompressible nematic liquid crystal system. J. Differ. Equations 258 (2015), 1519-1547. DOI 10.1016/j.jde.2014.11.002 | MR 3295591 | Zbl 1308.35222
  [24] M. Paicu, P. Zhang, Z. Zhang: Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density. Commun. Partial Differ. Equations 38 (2013), 1208-1234. DOI 10.1080/03605302.2013.780079 | MR 3169743 | Zbl 1314.35086
  [25] C. Wang: Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data. Arch. Ration. Mech. Anal. 200 (2011), 1-19. DOI 10.1007/s00205-010-0343-5 | MR 2781584 | Zbl 1285.35085
  [26] F. Xu, S. Hao, J. Yuan: Well-posedness for the density-dependent incompressible flow of liquid crystals. Math. Methods. Appl. Sci. 38 (2015), 2680-2702. DOI 10.1002/mma.3248 | MR 3382698 | Zbl 06523185
  [27] X. Xu, Z. Zhang: Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. J. Differ. Equations 252 (2012), 1169-1181. DOI 10.1016/j.jde.2011.08.028 | MR 2853534 | Zbl 1336.76005
  [28] J. Zhao, Q. Liu, S. Cui: Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Commun. Pure Appl. Anal. 12 (2013), 341-357. DOI 10.3934/cpaa.2013.12.341 | MR 2972434 | Zbl 1264.35007


Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer need to access the articles on their site, which is http://link.springer.com/journal/10587.

[Previous Article] [Next Article] [Contents of This Number] [Contents of Czechoslovak Mathematical Journal]