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November  2020, 25(11): 4515-4533. doi: 10.3934/dcdsb.2020110

A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow

Sili Liu , Xinhua Zhao and  Yingshan Chen ,

School of Mathematics, South China University of Technology, Guangzhou 510641, China

* Corresponding author: Yingshan Chen

Received  March 2019 Revised  October 2019 Published  March 2020

In this paper, we establish a new blowup criterion for the strong solution to the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in a bounded domain $ \Omega\subset\mathbb{R}^{3} $. Specifically, we obtain the blowup criterion in terms of $ \|P\|_{L^\infty_t BMO_{x}} $ and $ \|\nabla d\|_{L^s_t L^\infty_x} $, for any $ s>3 $. The appearance of vacuum could be allowed.

Keywords: Compressible nematic liquid crystal flows, strong solution, blowup criterion, BMO norm.
Mathematics Subject Classification: Primary: 35Q35, 35D35; Secondary: 76N10.
Citation: Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4515-4533. doi: 10.3934/dcdsb.2020110
References:
[1]

P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269.  doi: 10.1007/BF01759640.  Google Scholar

[2]

Y. Chen, X. Hou and L. Zhu, A new blowup criterion for strong solutions to the three-dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain, Math. Methods Appl. Sci., 40 (2017), 5526-5538.  doi: 10.1002/mma.4407.  Google Scholar

[3]

Y. Chen and M. Zhang, A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions, Kinet. Relat. Models, 9 (2016), 429-441.  doi: 10.3934/krm.2016001.  Google Scholar

[4]

S. Ding, J. Lin, C. Wang and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563.  doi: 10.3934/dcds.2012.32.539.  Google Scholar

[5]

S. Ding, C. Wang and H. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 357-371.  doi: 10.3934/dcdsb.2011.15.357.  Google Scholar

[6]

J. Gao, Q. Tao and Z.-A. Yao, Long-time behavior of solution for the compressible nematic liquid crystal flows in $\Bbb{R}^3$, J. Differential Equations, 261 (2016), 2334-2383.  doi: 10.1016/j.jde.2016.04.033.  Google Scholar

[7]

J. Gao, Q. Tao and Z.-A. Yao, A blowup criterion for the compressible nematic liquid crystal flows in dimension two, J. Math. Anal. Appl., 415 (2014), 33-52.  doi: 10.1016/j.jmaa.2014.01.039.  Google Scholar

[8]

T. Huang, C. Wang and H. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Differential Equations, 252 (2012), 2222-2265.  doi: 10.1016/j.jde.2011.07.036.  Google Scholar

[9]

T. Huang, C. Wang and H. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three, Arch. Ration. Mech. Anal., 204 (2012), 285-311.  doi: 10.1007/s00205-011-0476-1.  Google Scholar

[10]

X. Huang and Y. Wang, A Serrin criterion for compressible nematic liquid crystal flows, Math. Methods Appl. Sci., 36 (2013), 1363-1375.  doi: 10.1002/mma.2689.  Google Scholar

[11]

F. Jiang, S. Jiang and D. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 265 (2013), 3369-3397.  doi: 10.1016/j.jfa.2013.07.026.  Google Scholar

[12]

J. Lin, B. Lai and C. Wang, Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three, SIAM J. Math. Anal., 47 (2015), 2952-2983.  doi: 10.1137/15M1007665.  Google Scholar

[13]

S. Liu and S. Wang, A blow-up criterion for 2D compressible nematic liquid crystal flows in terms of density, Acta Appl. Math., 147 (2017), 39-62.  doi: 10.1007/s10440-016-0067-0.  Google Scholar

[14]

X.-G. Liu and J. Qing, Globally weak solutions to the flow of compressible liquid crystals system, Discrete Contin. Dyn. Syst., 33 (2013), 757-788.  doi: 10.3934/dcds.2013.33.757.  Google Scholar

[15]

Y. Liu, S. Zheng, H. Li and S. Liu, Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 37 (2017), 3921-3938.  doi: 10.3934/dcds.2017165.  Google Scholar

[16]

Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[17]

D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915.  doi: 10.1007/s00205-011-0488-x.  Google Scholar

show all references

References:
[1]

P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269.  doi: 10.1007/BF01759640.  Google Scholar

[2]

Y. Chen, X. Hou and L. Zhu, A new blowup criterion for strong solutions to the three-dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain, Math. Methods Appl. Sci., 40 (2017), 5526-5538.  doi: 10.1002/mma.4407.  Google Scholar

[3]

Y. Chen and M. Zhang, A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions, Kinet. Relat. Models, 9 (2016), 429-441.  doi: 10.3934/krm.2016001.  Google Scholar

[4]

S. Ding, J. Lin, C. Wang and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563.  doi: 10.3934/dcds.2012.32.539.  Google Scholar

[5]

S. Ding, C. Wang and H. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 357-371.  doi: 10.3934/dcdsb.2011.15.357.  Google Scholar

[6]

J. Gao, Q. Tao and Z.-A. Yao, Long-time behavior of solution for the compressible nematic liquid crystal flows in $\Bbb{R}^3$, J. Differential Equations, 261 (2016), 2334-2383.  doi: 10.1016/j.jde.2016.04.033.  Google Scholar

[7]

J. Gao, Q. Tao and Z.-A. Yao, A blowup criterion for the compressible nematic liquid crystal flows in dimension two, J. Math. Anal. Appl., 415 (2014), 33-52.  doi: 10.1016/j.jmaa.2014.01.039.  Google Scholar

[8]

T. Huang, C. Wang and H. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Differential Equations, 252 (2012), 2222-2265.  doi: 10.1016/j.jde.2011.07.036.  Google Scholar

[9]

T. Huang, C. Wang and H. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three, Arch. Ration. Mech. Anal., 204 (2012), 285-311.  doi: 10.1007/s00205-011-0476-1.  Google Scholar

[10]

X. Huang and Y. Wang, A Serrin criterion for compressible nematic liquid crystal flows, Math. Methods Appl. Sci., 36 (2013), 1363-1375.  doi: 10.1002/mma.2689.  Google Scholar

[11]

F. Jiang, S. Jiang and D. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 265 (2013), 3369-3397.  doi: 10.1016/j.jfa.2013.07.026.  Google Scholar

[12]

J. Lin, B. Lai and C. Wang, Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three, SIAM J. Math. Anal., 47 (2015), 2952-2983.  doi: 10.1137/15M1007665.  Google Scholar

[13]

S. Liu and S. Wang, A blow-up criterion for 2D compressible nematic liquid crystal flows in terms of density, Acta Appl. Math., 147 (2017), 39-62.  doi: 10.1007/s10440-016-0067-0.  Google Scholar

[14]

X.-G. Liu and J. Qing, Globally weak solutions to the flow of compressible liquid crystals system, Discrete Contin. Dyn. Syst., 33 (2013), 757-788.  doi: 10.3934/dcds.2013.33.757.  Google Scholar

[15]

Y. Liu, S. Zheng, H. Li and S. Liu, Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 37 (2017), 3921-3938.  doi: 10.3934/dcds.2017165.  Google Scholar

[16]

Y. Sun, C. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[17]

D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915.  doi: 10.1007/s00205-011-0488-x.  Google Scholar

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