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April  2015, 8(2): 323-340. doi: 10.3934/dcdss.2015.8.323

Diffusive transport in two-dimensional nematics

Ibrahim Fatkullin 1, and  Valeriy Slastikov 2,

1. 

Department of Mathematics, University of Arizona, Tucson, AZ 85721, United States
2. 

Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Received  June 2013 Revised  November 2013 Published  July 2014

We discuss a dynamical theory for nematic liquid crystals describing the stage of evolution in which the hydrodynamic fluid motion has already equilibrated and the subsequent evolution proceeds via diffusive motion of the orientational degrees of freedom. This diffusion induces a slow motion of singularities of the order parameter field. Using asymptotic methods for gradient flows, we establish a relation between the Doi-Smoluchowski kinetic equation and vortex dynamics in two-dimensional systems. We also discuss moment closures for the kinetic equation and Landau-de Gennes-type free energy dissipation.
Keywords: diffusive transport, Liquid crystals, Doi-Smoluchowski, nematics, vortex motion..
Mathematics Subject Classification: Primary: 76A15, 82B21, 82D30; Secondary: 35Q82, 35R0.
Citation: Ibrahim Fatkullin, Valeriy Slastikov. Diffusive transport in two-dimensional nematics. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 323-340. doi: 10.3934/dcdss.2015.8.323
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show all references

References:
[1]

J. Funct. Anal., 266 (2014), 4890-4907. doi: 10.1016/j.jfa.2014.01.024.  Google Scholar

[2]

Lectures in Mathematics, Birkhäuser Verlag, Basel-Boston-Berlin, 2005.  Google Scholar

[3]

Oxford University Press, 1994.  Google Scholar

[4]

Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser, Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0287-5.  Google Scholar

[5]

Clarendon Press, Oxford, 1995. Google Scholar

[6]

Clarendon Press, 1999. Google Scholar

[7]

Physica D, 77 (1994), 383-404. doi: 10.1016/0167-2789(94)90298-4.  Google Scholar

[8]

Methods and Applications of Analysis, 13 (2006), 181-198. doi: 10.4310/MAA.2006.v13.n2.a5.  Google Scholar

[9]

Journal of Rheology, 5 (1961), 23-34. doi: 10.1122/1.548883.  Google Scholar

[10]

Physica D, 237 (2008), 2577-2586. doi: 10.1016/j.physd.2008.03.048.  Google Scholar

[11]

Communications in Mathematical Sciences, 7 (2009), 917-938. doi: 10.4310/CMS.2009.v7.n4.a6.  Google Scholar

[12]

Archive for Rational Mechanics and Analysis, 28 (1968), 265-283. doi: 10.1007/BF00251810.  Google Scholar

[13]

Communications on Pure and Applied Mathematics, 49 (1996), 323-359. doi: 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E.  Google Scholar

[14]

Physica D, 43 (1990), 385-406. doi: 10.1016/0167-2789(90)90143-D.  Google Scholar

[15]

Journal of Functional Analysis, 152 (1998), 379-403. doi: 10.1006/jfan.1997.3170.  Google Scholar

[16]

Communications on Pure and Applied Mathematics, 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.  Google Scholar

[17]

Graduate Studies in Mathematics, AMS, 58, 1992. Google Scholar

[18]

arXiv:1206.5480, 2013. Google Scholar

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