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

Sen Ming; Han Yang; Zili Chen; Ls Yong

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 37-55
  • ISSN: 0011-4642

Abstract

top
The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space B ˙ p , 1 n / p - 1 ( ℝ n ) × B ˙ p , 1 n / p ( ℝ n ) with n < p < 2 n 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.

How to cite

top

Ming, Sen, et al. "The Cauchy problem for the liquid crystals system in the critical Besov space with negative index." Czechoslovak Mathematical Journal 67.1 (2017): 37-55. <http://eudml.org/doc/287923>.

@article{Ming2017,
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.},
author = {Ming, Sen, Yang, Han, Chen, Zili, Yong, Ls},
journal = {Czechoslovak Mathematical Journal},
keywords = {liquid crystals system; critical Besov space; negative index; well-posedness; blow-up},
language = {eng},
number = {1},
pages = {37-55},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Cauchy problem for the liquid crystals system in the critical Besov space with negative index},
url = {http://eudml.org/doc/287923},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Ming, Sen
AU - Yang, Han
AU - Chen, Zili
AU - Yong, Ls
TI - The Cauchy problem for the liquid crystals system in the critical Besov space with negative index
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 37
EP - 55
AB - 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.
LA - eng
KW - liquid crystals system; critical Besov space; negative index; well-posedness; blow-up
UR - http://eudml.org/doc/287923
ER -

References

top
  1. Abidi, H., 10.4171/RMI/505, Rev. Mat. Iberoam. 23 (2007), 537-586 French. (2007) Zbl1175.35099MR2371437DOI10.4171/RMI/505
  2. Abidi, H., Gui, G., Zhang, P., 10.1007/s00205-011-0473-4, Arch. Ration. Mech. Anal. 204 (2012), 189-230. (2012) Zbl1314.76021MR2898739DOI10.1007/s00205-011-0473-4
  3. Abidi, H., Zhang, P., On the global well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity, Available at Arxiv:1301.2371. MR3344050
  4. Bahouri, H., Chemin, J.-Y., Danchin, R., 10.1007/978-3-642-16830-7, Grundlehren der Mathematischen Wissenschaften 343, Springer, Heidelberg (2011). (2011) Zbl1227.35004MR2768550DOI10.1007/978-3-642-16830-7
  5. Cannone, M., 10.1016/s1874-5792(05)80006-0, Handbook of Mathematical Fluid Dynamics. Vol. III Elsevier/North Holland, Amsterdam S. Friedlander et al. (2004), 161-244. (2004) Zbl1222.35139MR2099035DOI10.1016/s1874-5792(05)80006-0
  6. Cavaterra, C., Rocca, E., Wu, H., 10.1016/j.jde.2013.03.009, J. Differ. Equations 255 (2013), 24-57. (2013) Zbl1282.35087MR3045633DOI10.1016/j.jde.2013.03.009
  7. Chen, Q., Miao, C., 10.1016/j.jde.2011.09.035, J. Differ. Equations 252 (2012), 2698-2724. (2012) Zbl1234.35193MR2860636DOI10.1016/j.jde.2011.09.035
  8. Danchin, R., 10.1081/PDE-100106132, Commun. Partial Differ. Equations 26 (2001), 1183-1233. (2001) Zbl1007.35071MR1855277DOI10.1081/PDE-100106132
  9. Danchin, R., Mucha, P. B., 10.1002/cpa.21409, Commun. Pure Appl. Math. 65 (2012), 1458-1480. (2012) Zbl1247.35088MR2957705DOI10.1002/cpa.21409
  10. Du, Y., Wang, K., 10.1016/j.jde.2013.07.066, J. Differ. Equations 256 (2014), 65-81. (2014) Zbl1320.35125MR3115835DOI10.1016/j.jde.2013.07.066
  11. Ericksen, J. L., 10.1007/bf00253358, Arch. Ration. Mech. Anal. 9 (1962), 371-378. (1962) Zbl0105.23403MR0137403DOI10.1007/bf00253358
  12. Fujita, H., Kato, T., 10.1007/BF00276188, Arch. Ration. Mech. Anal. 16 (1964), 269-315. (1964) Zbl0126.42301MR0166499DOI10.1007/BF00276188
  13. Hao, Y., Liu, X., 10.3934/cpaa.2014.13.225, Commun. Pure Appl. Anal. 13 (2014), 225-236. (2014) Zbl1273.76352MR3082558DOI10.3934/cpaa.2014.13.225
  14. Hong, M.-C., 10.1007/s00526-010-0331-5, Calc. Var. Partial Differ. Equ. 40 (2011), 15-36. (2011) Zbl1213.35014MR2745194DOI10.1007/s00526-010-0331-5
  15. Huang, J., Paicu, M., Zhang, P., 10.1016/j.matpur.2013.03.003, J. Math. Pures Appl. (9) 100 (2013), 806-831. (2013) Zbl1290.35184MR3125269DOI10.1016/j.matpur.2013.03.003
  16. Jiang, F., Jiang, S., Wang, D., 10.1007/s00205-014-0768-3, Arch. Ration. Mech. Anal. 214 (2014), 403-451. (2014) Zbl1307.35225MR3255696DOI10.1007/s00205-014-0768-3
  17. Li, X., Wang, D., 10.1016/j.jde.2011.08.045, J. Differ. Equations 252 (2012), 745-767. (2012) Zbl1277.35121MR2852225DOI10.1016/j.jde.2011.08.045
  18. Lin, F., 10.1002/cpa.3160420605, Commun. Pure Appl. Anal. 42 (1989), 789-814. (1989) Zbl0703.35173MR1003435DOI10.1002/cpa.3160420605
  19. Lin, F., Lin, J., Wang, C., 10.1007/s00205-009-0278-x, Arch. Ration. Mech. Anal. 197 (2010), 297-336. (2010) Zbl1346.76011MR2646822DOI10.1007/s00205-009-0278-x
  20. Lin, F., Liu, C., 10.3934/dcds.1996.2.1, Discrete Contin. Dyn. Syst. 2 (1996), 1-22. (1996) Zbl0948.35098MR1367385DOI10.3934/dcds.1996.2.1
  21. Lin, F., Liu, C., 10.1007/s002050000102, Arch. Ration. Mech. Anal. 154 (2000), 135-156. (2000) Zbl0963.35158MR1784963DOI10.1007/s002050000102
  22. Lin, J., Ding, S., 10.1002/mma.1548, Math. Methods Appl. Sci. 35 (2012), 158-173. (2012) Zbl1242.35006MR2876822DOI10.1002/mma.1548
  23. Liu, Q., Zhang, T., Zhao, J., 10.1016/j.jde.2014.11.002, J. Differ. Equations 258 (2015), 1519-1547. (2015) Zbl1308.35222MR3295591DOI10.1016/j.jde.2014.11.002
  24. Paicu, M., Zhang, P., Zhang, Z., 10.1080/03605302.2013.780079, Commun. Partial Differ. Equations 38 (2013), 1208-1234. (2013) Zbl1314.35086MR3169743DOI10.1080/03605302.2013.780079
  25. Wang, C., 10.1007/s00205-010-0343-5, Arch. Ration. Mech. Anal. 200 (2011), 1-19. (2011) Zbl1285.35085MR2781584DOI10.1007/s00205-010-0343-5
  26. Xu, F., Hao, S., Yuan, J., 10.1002/mma.3248, Math. Methods. Appl. Sci. 38 (2015), 2680-2702. (2015) Zbl06523185MR3382698DOI10.1002/mma.3248
  27. Xu, X., Zhang, Z., 10.1016/j.jde.2011.08.028, J. Differ. Equations 252 (2012), 1169-1181. (2012) Zbl1336.76005MR2853534DOI10.1016/j.jde.2011.08.028
  28. Zhao, J., Liu, Q., Cui, S., 10.3934/cpaa.2013.12.341, Commun. Pure Appl. Anal. 12 (2013), 341-357. (2013) Zbl1264.35007MR2972434DOI10.3934/cpaa.2013.12.341

NotesEmbed ?

top

You must be logged in to post comments.