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Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps
A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow
School of Mathematics, South China University of Technology, Guangzhou 510641, China |
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.
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P. Acquistapace,
On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269.
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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.
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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. |
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Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563.
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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