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August  2017, 37(7): 3921-3938. doi: 10.3934/dcds.2017165

Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows

Yang Liu 1, , Sining Zheng 1,, , Huapeng Li 2, and  Shengquan Liu 3,

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2. 

College of Science, Northeast Electric Power University, Jilin 132013, China
3. 

School of Mathematics, Liaoning University, Shenyang 110036, China

* Corresponding author: S. Zheng

Received  July 2015 Revised  February 2017 Published  April 2017

This paper studies the local existence of strong solutions to the Cauchy problem of the 2D simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows, coupled via $ρ$ (the density of the fluid), $u$ (the velocity of the field), and $d$ (the macroscopic/continuum molecular orientations). Notice that the technique used for the corresponding 3D local well-posedness of strong solutions fails treating the 2D case, because the $L^p$-norm ($p>2$) of the velocity $u$ cannot be controlled in terms only of $ρ^{\frac{1}{2}}u$ and $\nabla u$ here. In the present paper, under the framework of weighted approximation estimates introduced in [J. Li, Z. Liang, On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl. (2014) 640-671] for Navier-Stokes equations, we obtain the local existence of strong solutions to the 2D compressible nematic liquid crystal flows.

Keywords: Compressible nematic liquid crystal, vacuum, local strong solution, weighted estimates.
Mathematics Subject Classification: 76W05, 76N10, 35B45.
Citation: Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165
References:
[1]

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

[2]

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.

[3]

J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Ration. Mech. Anal., 9 (1962), 371-378.  doi: 10.1007/BF00253358.

[4]

P. Gennes de, The Physics of Liquid Crystals, Oxford, 1974.

[5]

P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 13 (2011), 137-146.  doi: 10.1007/s00021-009-0006-1.

[6]

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.

[7]

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

[8]

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.

[9]

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.

[10]

F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.

[11]

J. Li, Z. Xu and J. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows, preprint, arXiv: 1204.4966v1.

[12]

J. Li and Z. Liang, On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.

[13]

L. Li, Q. Liu and X. Zhong, Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum, preprint, arXiv: 1508.00235v1.

[14]

F. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.  doi: 10.1002/cpa.3160420605.

[15]

F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.

[16]

F. Lin and 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.

[17]

F. Lin and C. Wang, The Analysis of Harmonic Maps and Their Heat Flows, Hackensack: World Scientific Publishing Co. Pte. Ltd. , 2008. doi: 10.1142/9789812779533.

[18]

F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals Philos. Trans. R. Soc. Lond. Ser. A, 372 (2014), 20130361, 18 pp. doi: 10.1098/rsta.2013.0361.

[19]

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.

[20]

P. Lions, Mathematical Topics in Fluid Mechanics, vol. 1. Incompressible Models, NewYork: Oxford University Press, 1996.

[21]

Q. Liu, S. Liu, W. Tan and X. Zhong, Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differential Equations, 261 (2016), 6521-6569.  doi: 10.1016/j.jde.2016.08.044.

[22]

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.

[23]

S. Liu and J. Zhang, Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2631-2648.  doi: 10.3934/dcdsb.2016065.

[24]

S. Ma, Classical solutions for the compressible liquid crystal flows with nonnegative initial densities, J. Math. Anal. Appl., 397 (2013), 595-618.  doi: 10.1016/j.jmaa.2012.08.010.

[25]

T. Wang, A regularity condition of strong solutions to the two dimensional equations of compressible nematic liquid crystal flows, Math. Methods Appl. Sci., 40 (2017), 546-563.  doi: 10.1002/mma.3990.

show all references

References:
[1]

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

[2]

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.

[3]

J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Ration. Mech. Anal., 9 (1962), 371-378.  doi: 10.1007/BF00253358.

[4]

P. Gennes de, The Physics of Liquid Crystals, Oxford, 1974.

[5]

P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 13 (2011), 137-146.  doi: 10.1007/s00021-009-0006-1.

[6]

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.

[7]

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

[8]

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.

[9]

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.

[10]

F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.

[11]

J. Li, Z. Xu and J. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows, preprint, arXiv: 1204.4966v1.

[12]

J. Li and Z. Liang, On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.

[13]

L. Li, Q. Liu and X. Zhong, Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum, preprint, arXiv: 1508.00235v1.

[14]

F. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.  doi: 10.1002/cpa.3160420605.

[15]

F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.

[16]

F. Lin and 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.

[17]

F. Lin and C. Wang, The Analysis of Harmonic Maps and Their Heat Flows, Hackensack: World Scientific Publishing Co. Pte. Ltd. , 2008. doi: 10.1142/9789812779533.

[18]

F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals Philos. Trans. R. Soc. Lond. Ser. A, 372 (2014), 20130361, 18 pp. doi: 10.1098/rsta.2013.0361.

[19]

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.

[20]

P. Lions, Mathematical Topics in Fluid Mechanics, vol. 1. Incompressible Models, NewYork: Oxford University Press, 1996.

[21]

Q. Liu, S. Liu, W. Tan and X. Zhong, Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differential Equations, 261 (2016), 6521-6569.  doi: 10.1016/j.jde.2016.08.044.

[22]

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.

[23]

S. Liu and J. Zhang, Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2631-2648.  doi: 10.3934/dcdsb.2016065.

[24]

S. Ma, Classical solutions for the compressible liquid crystal flows with nonnegative initial densities, J. Math. Anal. Appl., 397 (2013), 595-618.  doi: 10.1016/j.jmaa.2012.08.010.

[25]

T. Wang, A regularity condition of strong solutions to the two dimensional equations of compressible nematic liquid crystal flows, Math. Methods Appl. Sci., 40 (2017), 546-563.  doi: 10.1002/mma.3990.

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