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July  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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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