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Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows
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 |
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.
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] | |
[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] | |
[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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