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Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities
Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions
1. | Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037 |
2. | College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 361000 |
References:
[1] |
M. A. Abdallah, F. Jiang and Z. Tan, Decay estimates for isentropic compressible magnetohydrodynamic equations in bounded domain, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 2211-2220.
doi: 10.1016/S0252-9602(12)60171-4. |
[2] |
S. J. Ding, J. R. Huang, F. G. Xia, H. Y. Wen and R. Z. Zi, Incompressible limit of the compressible nematic liquid crystal flow, J. Funct. Anal., 264 (2013), 1711-1756.
doi: 10.1016/j.jfa.2013.01.011. |
[3] |
E. Feireisl and H. Petzeltová, Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96.
doi: 10.1007/s002050050181. |
[4] |
M. Grasselli and H. Wu, Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal., 45 (2013), 965-1002.
doi: 10.1137/120866476. |
[5] |
J. L. Hineman and C. Y. Wang, Well-posedness of Nematic liquid crystal flow in $L_{u l o c}^3(R^3)$, Arch. Ration. Mech. Anal., 210 (2013), 177-218.
doi: 10.1007/s00205-013-0643-7. |
[6] |
M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.
doi: 10.1007/s00526-010-0331-5. |
[7] |
X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Analysis, 252 (2013), 2678-2699.
doi: 10.1137/120898814. |
[8] |
X. P. Hu and H. Wu, Long-time dynamics of the nonhomogeneous incompressible flow of nematic liquid crystals, Commun. Math. Sci., 11 (2013), 779-806.
doi: 10.4310/CMS.2013.v11.n3.a6. |
[9] |
T. Huang, C. Y. Wang and H. Y. 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] |
T. Huang, C. Y. Wang and H. Y. 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. |
[11] |
F. Jiang, S. Jiang and D. H. 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] |
F. Jiang, S. Jiang, and D. H. Wang, Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions, Arch. Ration. Mech. Anal., 214 (2014), 403-451.
doi: 10.1007/s00205-014-0768-3. |
[13] |
F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266.
doi: 10.1002/mma.1132. |
[14] |
F. Jiang and Z. Tan, On the domain dependence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Methods Appl. Sci., 32 (2009), 2350-2367.
doi: 10.1002/mma.1138. |
[15] |
J. Li, Z. H. Xu and J. W. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows, preprint, arXiv:1204.4966. |
[16] |
J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80-94.
doi: 10.1016/j.na.2013.12.023. |
[17] |
X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767.
doi: 10.1016/j.jde.2011.08.045. |
[18] |
F. H. 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. |
[19] |
F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[20] |
F. H. 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. |
[21] |
F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Cont. Dyn. S., 2 (1996), 1-22. |
[22] |
F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, preprint, arXiv:1408.4146. |
[23] |
F. H. Lin and C. Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp. |
[24] |
J. Y. Lin, B. S. Lai and C. Y. 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. |
[25] |
P. Lions, Mathematical Topics in Fluid Mechanics: Incompressible models, Oxford University Press, New York, 1996. |
[26] |
Q. Liu, On the temporal decay of solutions to the two-dimensional nematic liquid crystal flows, preprint, arXiv:1409.8475. |
[27] |
X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math. Ser. A, 30 (2009), 1-20. |
[28] |
D. G. Matteis and G. E. Virga, Director libration in nematoacoustics, Physical Review E, 83 (2011), 011703.
doi: 10.1103/PhysRevE.83.011703. |
[29] |
A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004. |
[30] |
M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222.
doi: 10.1007/BF00752111. |
[31] |
L. Simon, Asymptotics for a class of nonlinear evolution equation, with applications to geometri problems, Ann. of Math.(2), 118 (1983), 525-571.
doi: 10.2307/2006981. |
[32] |
C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal., 200 (2011), 1-19.
doi: 10.1007/s00205-010-0343-5. |
[33] |
C. Y. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $\mathbbS^2$, J. Funct. Anal., 266 (2014), 5360-5376.
doi: 10.1016/j.jfa.2014.02.023. |
[34] |
D. H. 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. |
[35] |
H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531.
doi: 10.1016/j.nonrwa.2010.10.010. |
[36] |
H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396.
doi: 10.3934/dcds.2010.26.379. |
[37] |
H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345.
doi: 10.1007/s00526-011-0460-5. |
[38] |
Y. Zhou, J. S. Fan and G. Nakamura, Global strong solution to the density-dependent 2-D liquid crystal flows, Abstr. Appl. Anal., Art. ID 947291 (2013), 5pp. |
show all references
References:
[1] |
M. A. Abdallah, F. Jiang and Z. Tan, Decay estimates for isentropic compressible magnetohydrodynamic equations in bounded domain, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 2211-2220.
doi: 10.1016/S0252-9602(12)60171-4. |
[2] |
S. J. Ding, J. R. Huang, F. G. Xia, H. Y. Wen and R. Z. Zi, Incompressible limit of the compressible nematic liquid crystal flow, J. Funct. Anal., 264 (2013), 1711-1756.
doi: 10.1016/j.jfa.2013.01.011. |
[3] |
E. Feireisl and H. Petzeltová, Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96.
doi: 10.1007/s002050050181. |
[4] |
M. Grasselli and H. Wu, Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal., 45 (2013), 965-1002.
doi: 10.1137/120866476. |
[5] |
J. L. Hineman and C. Y. Wang, Well-posedness of Nematic liquid crystal flow in $L_{u l o c}^3(R^3)$, Arch. Ration. Mech. Anal., 210 (2013), 177-218.
doi: 10.1007/s00205-013-0643-7. |
[6] |
M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.
doi: 10.1007/s00526-010-0331-5. |
[7] |
X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Analysis, 252 (2013), 2678-2699.
doi: 10.1137/120898814. |
[8] |
X. P. Hu and H. Wu, Long-time dynamics of the nonhomogeneous incompressible flow of nematic liquid crystals, Commun. Math. Sci., 11 (2013), 779-806.
doi: 10.4310/CMS.2013.v11.n3.a6. |
[9] |
T. Huang, C. Y. Wang and H. Y. 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] |
T. Huang, C. Y. Wang and H. Y. 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. |
[11] |
F. Jiang, S. Jiang and D. H. 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] |
F. Jiang, S. Jiang, and D. H. Wang, Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions, Arch. Ration. Mech. Anal., 214 (2014), 403-451.
doi: 10.1007/s00205-014-0768-3. |
[13] |
F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266.
doi: 10.1002/mma.1132. |
[14] |
F. Jiang and Z. Tan, On the domain dependence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Methods Appl. Sci., 32 (2009), 2350-2367.
doi: 10.1002/mma.1138. |
[15] |
J. Li, Z. H. Xu and J. W. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows, preprint, arXiv:1204.4966. |
[16] |
J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80-94.
doi: 10.1016/j.na.2013.12.023. |
[17] |
X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767.
doi: 10.1016/j.jde.2011.08.045. |
[18] |
F. H. 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. |
[19] |
F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[20] |
F. H. 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. |
[21] |
F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Cont. Dyn. S., 2 (1996), 1-22. |
[22] |
F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, preprint, arXiv:1408.4146. |
[23] |
F. H. Lin and C. Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp. |
[24] |
J. Y. Lin, B. S. Lai and C. Y. 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. |
[25] |
P. Lions, Mathematical Topics in Fluid Mechanics: Incompressible models, Oxford University Press, New York, 1996. |
[26] |
Q. Liu, On the temporal decay of solutions to the two-dimensional nematic liquid crystal flows, preprint, arXiv:1409.8475. |
[27] |
X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math. Ser. A, 30 (2009), 1-20. |
[28] |
D. G. Matteis and G. E. Virga, Director libration in nematoacoustics, Physical Review E, 83 (2011), 011703.
doi: 10.1103/PhysRevE.83.011703. |
[29] |
A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004. |
[30] |
M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222.
doi: 10.1007/BF00752111. |
[31] |
L. Simon, Asymptotics for a class of nonlinear evolution equation, with applications to geometri problems, Ann. of Math.(2), 118 (1983), 525-571.
doi: 10.2307/2006981. |
[32] |
C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal., 200 (2011), 1-19.
doi: 10.1007/s00205-010-0343-5. |
[33] |
C. Y. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $\mathbbS^2$, J. Funct. Anal., 266 (2014), 5360-5376.
doi: 10.1016/j.jfa.2014.02.023. |
[34] |
D. H. 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. |
[35] |
H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531.
doi: 10.1016/j.nonrwa.2010.10.010. |
[36] |
H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396.
doi: 10.3934/dcds.2010.26.379. |
[37] |
H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345.
doi: 10.1007/s00526-011-0460-5. |
[38] |
Y. Zhou, J. S. Fan and G. Nakamura, Global strong solution to the density-dependent 2-D liquid crystal flows, Abstr. Appl. Anal., Art. ID 947291 (2013), 5pp. |
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