×

Liquid crystal flows in two dimensions. (English) Zbl 1346.76011

Summary: This paper is concerned with a simplified hydrodynamic equation, proposed by J. L. Ericksen [Arch. Ration. Mech. Anal. 9, 371–378 (1962; Zbl 0105.23403)] and F. M. Leslie [ibid. 28, 265–283 (1968; Zbl 0159.57101)], modeling the flow of nematic liquid crystals. In dimension two, we establish both interior and boundary regularity theorems for such a flow under smallness conditions. As a consequence, we establish the existence of global (in time) weak solutions on a bounded smooth domain in \({\mathbb{R}^2}\) which are smooth everywhere with possible exceptions of finitely many singular times.

MSC:

76A15 Liquid crystals
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
35Q35 PDEs in connection with fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chang K.C.: Heat flow and boundary value problem for harmonic maps. Annales de l’institut Henri Poincaré (C) Analyse non linéaire 6(5), 363–395 (1989)
[2] Chang K.C., Ding W.Y., Ye R.: Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differ. Geom. 36, 507–515 (1992) · Zbl 0765.53026
[3] Constantin, P., Seregin, S.: Hölder continuity of solutions of 2D Navier-Stokes equations with singular forcing. Preprint, 2009 · Zbl 1223.35099
[4] Chen Y.M., Lin F.H.: Evolution of harmonic maps with Dirichlet boundary conditions. Comm. Anal. Geom. 1(3–4), 327–346 (1993) · Zbl 0845.35049
[5] Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of Navier-Stokes equations. CPAM 35, 771–831 (1982) · Zbl 0509.35067
[6] de Gennes, P.G.: The Physics of Liquid Crystals. Oxford, 1974 · Zbl 0295.76005
[7] Ericksen J.L.: Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9, 371–378 (1962) · Zbl 0105.23403
[8] Hong, M.C.: Global existence of solutions of the simplified Ericksen-Leslie system in \({\mathbb{R}^2}\) . Preprint
[9] Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incompressible Fluid, Gordon and Breach, New York, 1969 · Zbl 0184.52603
[10] Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasilinear equations of parabolic type. Am. Math. Soc., Providence, 1968
[11] Leslie F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1962) · Zbl 0159.57101
[12] Lemaire L.: Applications harmoniques de surfaces riemanniennes. J. Differ. Geom. 13(1), 51–78 (1978) · Zbl 0388.58003
[13] Lin F.H.: A new proof of the Caffarelli-Kohn-Nirenberg Theorem. Comm. Pure. Appl. Math. LI, 0241–0257 (1998) · Zbl 0958.35102 · doi:10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A
[14] Lin F.H., Liu C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. CPAM XLVIII, 501–537 (1995) · Zbl 0842.35084
[15] Lin F.H., Liu C.: Partial regularity of the dynamic system modeling the flow of liquid crystals. DCDS 2(1), 1–22 (1998) · Zbl 0948.35098
[16] Lin F.H., Wang C.Y.: Harmonic and quasi-harmonic spheres. II. Comm. Anal. Geom. 10(2), 341–375 (2002) · Zbl 1042.58005
[17] Qing J.: On singularities of the heat flow for harmonic maps from surfaces into spheres. Comm. Anal. Geom. 3(1–2), 297–315 (1995) · Zbl 0868.58021
[18] Seregin G., Shilkin T., Solonnikov V.: Boundary partial regularity for the Navier-Stokes equations. J. Math. Sci. 132(3), 339–358 (2006) · Zbl 1095.35031 · doi:10.1007/s10958-005-0502-7
[19] Solonnikov, V.A.: On Schauder estimates for the evolution generalized stokes problem. Hyperbolic Problems and Regularity Questions, Trends in Mathematics, Birkhäuser, Basel, 197–205, 2007 · Zbl 1123.35005
[20] Solonnikov V.A.: L p -Estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain. J. Math. Sci. 105(5), 2448–2484 (2001) · doi:10.1023/A:1011321430954
[21] Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. (2) 113(1), 1–24 (1981) · Zbl 0462.58014
[22] Schoen, R., Uhlenbeck, K.: Approximation of Sobolev maps between Riemannian manifolds. Preprint (1984) · Zbl 0555.58011
[23] Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helvetici 60, 558–581 (1985) · Zbl 0595.58013 · doi:10.1007/BF02567432
[24] Temam, R.: Navier-Stokes equations. Studies in Mathematics and its Applications, Vol. 2, North Holland, Amsterdam, 1977 · Zbl 0383.35057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.