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Structure formation in sheared polymer-rod nanocomposites
Scaling invariant blow-up criteria for simplified versions of Ericksen-Leslie system
| 1. | Department of Mathematics, Chung-Ang University, Seoul 156-756, South Korea |
References:
| [1] |
T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equation, Commun. Math. Phys., 94 (1984), 61-66.
doi: 10.1007/BF01212349. |
| [2] |
B. da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbb{R}^{N}$, Chinese Ann. of Math., 16 (1995), 407-412. |
| [3] |
Q. Chen, A. Tan and G. Wu, LPS's criterion for incompressible nematic liquid crystal flows, Acta Mathematica Scientia, 34 (2014), 1072-1080.
doi: 10.1016/S0252-9602(14)60070-9. |
| [4] |
J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378. |
| [5] |
E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231. |
| [6] |
M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in $\mathbb{R}^2$, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.
doi: 10.1007/s00526-010-0331-5. |
| [7] |
M. C. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbb{R}^2$, Adv. Math., 231 (2012), 1364-1400.
doi: 10.1016/j.aim.2012.06.009. |
| [8] |
T. Huang and C. Wang, Blow-up criterion for nematic liquid crystal flows, Commun. Partial Diff. Equations, 37 (2012), 875-884.
doi: 10.1080/03605302.2012.659366. |
| [9] |
J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta. Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
| [10] |
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
| [11] |
F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501-537.
doi: 10.1002/cpa.3160480503. |
| [12] |
F. H. Lin, J. 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. |
| [13] |
Q. Liu and J. Zhao, Logarithmical blow-up criteria for the nematic liquid crystal flows, Nonlinear Anal. Real World Appl., 16 (2014), 178-190.
doi: 10.1016/j.nonrwa.2013.09.017. |
| [14] |
T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., 34 (2003), 1318-1330.
doi: 10.1137/S0036141001395868. |
| [15] |
J. Serrin, The initial value problem for the Navier-Stokes equations, in Nonlinear Probl., Proc. Sympos. Madison 1962 (ed. R. Lamger), Univ. Wisconsin Press, 1963, 69-98. |
| [16] |
X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181.
doi: 10.1016/j.jde.2011.08.028. |
show all references
References:
| [1] |
T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equation, Commun. Math. Phys., 94 (1984), 61-66.
doi: 10.1007/BF01212349. |
| [2] |
B. da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbb{R}^{N}$, Chinese Ann. of Math., 16 (1995), 407-412. |
| [3] |
Q. Chen, A. Tan and G. Wu, LPS's criterion for incompressible nematic liquid crystal flows, Acta Mathematica Scientia, 34 (2014), 1072-1080.
doi: 10.1016/S0252-9602(14)60070-9. |
| [4] |
J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378. |
| [5] |
E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231. |
| [6] |
M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in $\mathbb{R}^2$, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.
doi: 10.1007/s00526-010-0331-5. |
| [7] |
M. C. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbb{R}^2$, Adv. Math., 231 (2012), 1364-1400.
doi: 10.1016/j.aim.2012.06.009. |
| [8] |
T. Huang and C. Wang, Blow-up criterion for nematic liquid crystal flows, Commun. Partial Diff. Equations, 37 (2012), 875-884.
doi: 10.1080/03605302.2012.659366. |
| [9] |
J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta. Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
| [10] |
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
| [11] |
F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501-537.
doi: 10.1002/cpa.3160480503. |
| [12] |
F. H. Lin, J. 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. |
| [13] |
Q. Liu and J. Zhao, Logarithmical blow-up criteria for the nematic liquid crystal flows, Nonlinear Anal. Real World Appl., 16 (2014), 178-190.
doi: 10.1016/j.nonrwa.2013.09.017. |
| [14] |
T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., 34 (2003), 1318-1330.
doi: 10.1137/S0036141001395868. |
| [15] |
J. Serrin, The initial value problem for the Navier-Stokes equations, in Nonlinear Probl., Proc. Sympos. Madison 1962 (ed. R. Lamger), Univ. Wisconsin Press, 1963, 69-98. |
| [16] |
X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181.
doi: 10.1016/j.jde.2011.08.028. |
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