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April  2015, 8(2): 381-388. doi: 10.3934/dcdss.2015.8.381

Scaling invariant blow-up criteria for simplified versions of Ericksen-Leslie system

Jihoon Lee 1,

1. 

Department of Mathematics, Chung-Ang University, Seoul 156-756, South Korea

Received  December 2012 Revised  November 2013 Published  July 2014

In this paper, we establish scaling invariant blow-up criteria for a classical solution to simplified version of two and three dimensional Ericksen-Leslie system. We also consider the model replacing the Navier-Stokes equations by Stokes equations in the system and obtain blow-up criterion in three dimensions.
Keywords: blow-up criterion, Navier-Stokes., Ericksen-Leslie system.
Mathematics Subject Classification: Primary: 35B44, 35D30; Secondary: 76A1.
Citation: Jihoon Lee. Scaling invariant blow-up criteria for simplified versions of Ericksen-Leslie system. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 381-388. doi: 10.3934/dcdss.2015.8.381
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.  Google Scholar

[2]

B. da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbbR^n$, Chinese Ann. of Math., 16 (1995), 407-412.  Google Scholar

[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.  Google Scholar

[4]

J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378.  Google Scholar

[5]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.  Google Scholar

[6]

M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in $\mathbbR^2$, Calc. Var. Partial Differential Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5.  Google Scholar

[7]

M. C. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbbR^2$, Adv. Math., 231 (2012), 1364-1400. doi: 10.1016/j.aim.2012.06.009.  Google Scholar

[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.  Google Scholar

[9]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta. Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

B. da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbbR^n$, Chinese Ann. of Math., 16 (1995), 407-412.  Google Scholar

[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.  Google Scholar

[4]

J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378.  Google Scholar

[5]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.  Google Scholar

[6]

M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in $\mathbbR^2$, Calc. Var. Partial Differential Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5.  Google Scholar

[7]

M. C. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbbR^2$, Adv. Math., 231 (2012), 1364-1400. doi: 10.1016/j.aim.2012.06.009.  Google Scholar

[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.  Google Scholar

[9]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta. Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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