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A Serrin criterion for compressible nematic liquid crystal flows. (English) Zbl 1291.35220
Summary: This paper is concerned with a simplified system, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. We establish a blowup criterion for three-dimensional compressible nematic liquid crystal flows, which is analogous to the well-known Serrin’s blowup criterion for three-dimensional incompressible viscous flows.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76A15 Liquid crystals
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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References:
[1] Ericksen, Hydrostatic theory of liquid crystal, Archive Rational Mechanics and Analysis 9 pp 371– (1962) · Zbl 0105.23403
[2] Leslie, Some constitutive equations for liquid crystals, Archive Rational Mechanics and Analysis 28 pp 265– (1968) · Zbl 0159.57101
[3] Lin, Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena, Communications on Pure and Applied Mathematics 42 pp 789– (1989) · Zbl 0703.35173
[4] Lin, Liquid crystal flows in two dimensions, Archive Rational Mechanics and Analysis 197 pp 297– (2010) · Zbl 1346.76011
[5] Lin, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Continuous Dynamical Systems 2 (1) pp 122– (1996) · Zbl 0948.35098
[6] Feireisl, Dynamics of Viscous Compressible Fluids (2004) · Zbl 1080.76001
[7] Lions, Mathematical Topics in Fluid Mechanics 2 (1998)
[8] Cho, Unique solvability of the initial boundary value problems for compressible viscous fluids, Journal de Mathématiques Pures et Appliquées 83 pp 243– (2004) · Zbl 1080.35066
[9] Choe, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, Journal of Differential Equations 190 pp 504– (2003) · Zbl 1022.35037
[10] Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général, Bulletin de la Société Mathématique de France 90 pp 487– (1962) · Zbl 0113.19405
[11] Serrin, On the uniqueness of compressible fluid motion, Archive Rational Mechanics and Analysis 3 pp 271– (1959) · Zbl 0089.19103
[12] Choe, Regularity of weak solutions of the compressible Navier-Stokes equations, Journal of the Korean Mathematical Society 40 pp 1031– (2003) · Zbl 1034.76049
[13] Fan, Blow-up criteria for the Navier-Stokes equations of compressible fluids, Journal of Hyperbolic Differential Equations 5 pp 167– (2008) · Zbl 1142.76049
[14] Huang XD Some results on blowup of solutions to the compressible Navier-Stokes equations PhD Thesis 2009
[15] Huang, A blow-up criterion for classical solutions to the compressible Navier-Stokes equations, Science China Mathematics 53 (3) pp 671– (2010) · Zbl 1256.35059
[16] Beale, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Communication on Mathematical Physics 94 pp 61– (1984) · Zbl 0573.76029
[17] Huang, Blowup criterion for the three-dimensional viscous barotropic flows with vacuum states, Communication on Mathematical Physics 301 pp 23– (2011) · Zbl 1213.35135
[18] Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Archive Rational Mechanics and Analysis 9 pp 187– (1962) · Zbl 0106.18302
[19] Huang, Serrin type criterion for the three-dimensional viscous compressible flows, SIAM Journal on Mathematical Analysis 43 pp 1872– (2011) · Zbl 1241.35161
[20] Sun, A Beale-Kato-Majda blow-up criterion for the compressible Navier-Stokes equations, Journal de Mathématiques Pures et Appliquées 95 pp 36– (2011) · Zbl 1205.35212
[21] Ding, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Continuous Dynamical Systems 15 (2) pp 357– (2011)
[22] Huang T Wang CY Wen HY Strong solutions of the compressible nematic liquid crystal flow
[23] Huang T Wang CY Wen HY Blow up criterion for compressible nematic liquid crystal flows in dimension three · Zbl 1314.76010
[24] Liu XG Liu LM A blow-up criterion for the compressible liquid crystal system
[25] Wang, Heat flow of harmonic maps whose gradients belong to LxnLt, Archive Rational Mechanics and Analysis 188 pp 309– (2008) · Zbl 1156.35052
[26] Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, Journal of Differential Equations 120 (1) pp 215– (1995) · Zbl 0836.35120
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