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A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$
Well-posedness for the three-dimensional compressible liquid crystal flows
1. | College of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China |
2. | Institute of Applied Physics & Computational Math., Beijing 100088 |
References:
[1] |
K. C. Chang, W. Y. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces,, J. Diff. Geom., 36 (1992), 507.
|
[2] |
H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, J. Diff. Equations, 190 (2003), 504.
doi: 10.1016/S0022-0396(03)00015-9. |
[3] |
Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value prob lems for compressible viscous fluids,, J. Math. Pures Appl., 83 (2004), 243.
doi: 10.1016/j.matpur.2003.11.004. |
[4] |
Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum,, J. Diff. Equations, 228 (2006), 377.
doi: 10.1016/j.jde.2006.05.001. |
[5] |
S. Ding, C. Wang and H. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one,, Discrete Conti. Dyna. Sys. Ser. B, 15 (2011), 357.
doi: 10.3934/dcdsb.2011.15.357. |
[6] |
S. Ding, J. Lin, C. Wang and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D,, Discrete Conti. Dyna. Sys., 32 (2012), 539.
doi: 10.3934/dcds.2012.32.539. |
[7] |
J. Ericksen, Conservation laws for liquid crystals,, Trans. Soc. Rheol., 5 (1961), 22.
doi: 10.1122/1.548883. |
[8] |
J. Ericksen, Equilibrium theory for liquid crystals, in: G. Brown (Ed.),, Advances in Liquid Crystals, 2 (1976), 233.
doi: 10.1016/B978-0-12-025002-8.50012-9. |
[9] |
J. Ericksen, Continuum theory of nematic liquid crystals,, Molecular Crystals, 7 (2007), 153.
doi: 10.1080/15421406908084869. |
[10] |
M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two,, Calc. Var. Partial Diff. Equations, 40 (2011), 15.
doi: 10.1007/s00526-010-0331-5. |
[11] |
X. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals,, SIAM J. Math. Anal., 45 (2013), 2678.
doi: 10.1137/120898814. |
[12] |
T. Huang, C. Wang and H. Wen, Strong solutions of the compressible nematic liquid crystal flow,, J. Diff. Equations, 252 (2012), 2222.
doi: 10.1016/j.jde.2011.07.036. |
[13] |
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.
doi: 10.1007/s00205-011-0476-1. |
[14] |
F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system,, Math. Meth. Appl. Sci., 32 (2009), 2243.
doi: 10.1002/mma.1132. |
[15] |
F. M. Leslie, Theory of flow phenomena in liquid crystals, in: G. Brown (Ed.),, Advances in Liquid Crystals, 4 (1979), 1.
doi: 10.1016/B978-0-12-025004-2.50008-9. |
[16] |
X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals,, Trans. Amer. Math. Soc., 367 (2015), 2301.
doi: 10.1090/S0002-9947-2014-05924-2. |
[17] |
F. H. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789.
doi: 10.1002/cpa.3160420605. |
[18] |
F. H. Lin, Existence of solutions for the Ericksen-Leslie system,, Arch. Rat. Mech. Anal., 154 (2000), 135.
doi: 10.1007/s002050000102. |
[19] |
F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501.
doi: 10.1002/cpa.3160480503. |
[20] |
F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals,, Discrete Conti. Dyna. Sys., 2 (1996), 1.
|
[21] |
F. H. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions,, Arch. Ration. Mech. Anal., 197 (2010), 297.
doi: 10.1007/s00205-009-0278-x. |
[22] |
C. Liu, Dynamic theory for incompressible smectic-A liquid crystals,, Discrete Conti. Dyna. Sys., 6 (2000), 591.
doi: 10.3934/dcds.2000.6.591. |
[23] |
X. Liu, L. Liu and Y. Hao, Existence of strong solutions for the compressible Ericksen-Leslie model,, , (). Google Scholar |
[24] |
X. Liu and Z. Zhang, {Existence of the flow of liquid crystals system,, Chinese Ann. Math., 30 (2009), 1.
|
[25] |
S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds,, Comm. Partial Diff. Equations, 27 (2001), 1103.
doi: 10.1081/PDE-120004895. |
[26] |
C. 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.
doi: 10.1007/s00205-010-0343-5. |
[27] |
H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals,, Nonlinear Analysis: Real World Applications, 12 (2011), 1510.
doi: 10.1016/j.nonrwa.2010.10.010. |
show all references
References:
[1] |
K. C. Chang, W. Y. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces,, J. Diff. Geom., 36 (1992), 507.
|
[2] |
H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, J. Diff. Equations, 190 (2003), 504.
doi: 10.1016/S0022-0396(03)00015-9. |
[3] |
Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value prob lems for compressible viscous fluids,, J. Math. Pures Appl., 83 (2004), 243.
doi: 10.1016/j.matpur.2003.11.004. |
[4] |
Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum,, J. Diff. Equations, 228 (2006), 377.
doi: 10.1016/j.jde.2006.05.001. |
[5] |
S. Ding, C. Wang and H. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one,, Discrete Conti. Dyna. Sys. Ser. B, 15 (2011), 357.
doi: 10.3934/dcdsb.2011.15.357. |
[6] |
S. Ding, J. Lin, C. Wang and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D,, Discrete Conti. Dyna. Sys., 32 (2012), 539.
doi: 10.3934/dcds.2012.32.539. |
[7] |
J. Ericksen, Conservation laws for liquid crystals,, Trans. Soc. Rheol., 5 (1961), 22.
doi: 10.1122/1.548883. |
[8] |
J. Ericksen, Equilibrium theory for liquid crystals, in: G. Brown (Ed.),, Advances in Liquid Crystals, 2 (1976), 233.
doi: 10.1016/B978-0-12-025002-8.50012-9. |
[9] |
J. Ericksen, Continuum theory of nematic liquid crystals,, Molecular Crystals, 7 (2007), 153.
doi: 10.1080/15421406908084869. |
[10] |
M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two,, Calc. Var. Partial Diff. Equations, 40 (2011), 15.
doi: 10.1007/s00526-010-0331-5. |
[11] |
X. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals,, SIAM J. Math. Anal., 45 (2013), 2678.
doi: 10.1137/120898814. |
[12] |
T. Huang, C. Wang and H. Wen, Strong solutions of the compressible nematic liquid crystal flow,, J. Diff. Equations, 252 (2012), 2222.
doi: 10.1016/j.jde.2011.07.036. |
[13] |
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.
doi: 10.1007/s00205-011-0476-1. |
[14] |
F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system,, Math. Meth. Appl. Sci., 32 (2009), 2243.
doi: 10.1002/mma.1132. |
[15] |
F. M. Leslie, Theory of flow phenomena in liquid crystals, in: G. Brown (Ed.),, Advances in Liquid Crystals, 4 (1979), 1.
doi: 10.1016/B978-0-12-025004-2.50008-9. |
[16] |
X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals,, Trans. Amer. Math. Soc., 367 (2015), 2301.
doi: 10.1090/S0002-9947-2014-05924-2. |
[17] |
F. H. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789.
doi: 10.1002/cpa.3160420605. |
[18] |
F. H. Lin, Existence of solutions for the Ericksen-Leslie system,, Arch. Rat. Mech. Anal., 154 (2000), 135.
doi: 10.1007/s002050000102. |
[19] |
F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501.
doi: 10.1002/cpa.3160480503. |
[20] |
F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals,, Discrete Conti. Dyna. Sys., 2 (1996), 1.
|
[21] |
F. H. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions,, Arch. Ration. Mech. Anal., 197 (2010), 297.
doi: 10.1007/s00205-009-0278-x. |
[22] |
C. Liu, Dynamic theory for incompressible smectic-A liquid crystals,, Discrete Conti. Dyna. Sys., 6 (2000), 591.
doi: 10.3934/dcds.2000.6.591. |
[23] |
X. Liu, L. Liu and Y. Hao, Existence of strong solutions for the compressible Ericksen-Leslie model,, , (). Google Scholar |
[24] |
X. Liu and Z. Zhang, {Existence of the flow of liquid crystals system,, Chinese Ann. Math., 30 (2009), 1.
|
[25] |
S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds,, Comm. Partial Diff. Equations, 27 (2001), 1103.
doi: 10.1081/PDE-120004895. |
[26] |
C. 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.
doi: 10.1007/s00205-010-0343-5. |
[27] |
H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals,, Nonlinear Analysis: Real World Applications, 12 (2011), 1510.
doi: 10.1016/j.nonrwa.2010.10.010. |
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