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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
References:
[1]

R. Alicandro and M. Ponsiglione, Ginzburg-Landau functionals and renormalized energy: A revised $\Gamma$-convergence approach, J. Funct. Anal., 266 (2014), 4890-4907. doi: 10.1016/j.jfa.2014.01.024.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser Verlag, Basel-Boston-Berlin, 2005.  Google Scholar

[3]

A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, 1994.  Google Scholar

[4]

F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices, 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]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1995. Google Scholar

[6]

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, 1999. Google Scholar

[7]

W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Physica D, 77 (1994), 383-404. doi: 10.1016/0167-2789(94)90298-4.  Google Scholar

[8]

W. E and P. Zhang, A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit, Methods and Applications of Analysis, 13 (2006), 181-198. doi: 10.4310/MAA.2006.v13.n2.a5.  Google Scholar

[9]

J. L. Ericksen, Conservation laws for liquid crystals, Journal of Rheology, 5 (1961), 23-34. doi: 10.1122/1.548883.  Google Scholar

[10]

I. Fatkullin and V. Slastikov, On spatial variations of nematic ordering, Physica D, 237 (2008), 2577-2586. doi: 10.1016/j.physd.2008.03.048.  Google Scholar

[11]

I. Fatkullin and V. Slastikov, Vortices in two-dimensional nematics, Communications in Mathematical Sciences, 7 (2009), 917-938. doi: 10.4310/CMS.2009.v7.n4.a6.  Google Scholar

[12]

F. M. Leslie, Some constitutive equations for liquid crystals, Archive for Rational Mechanics and Analysis, 28 (1968), 265-283. doi: 10.1007/BF00251810.  Google Scholar

[13]

F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices, 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]

J. C. Neu, Vortices in complex scalar fields, Physica D, 43 (1990), 385-406. doi: 10.1016/0167-2789(90)90143-D.  Google Scholar

[15]

E. Sandier, Lower bounds for the energy of unit vector fields and applications, Journal of Functional Analysis, 152 (1998), 379-403. doi: 10.1006/jfan.1997.3170.  Google Scholar

[16]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Communications on Pure and Applied Mathematics, 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.  Google Scholar

[17]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, AMS, 58, 1992. Google Scholar

[18]

P. Zhang W. Wang and Z. Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation, arXiv:1206.5480, 2013. Google Scholar

show all references

References:
[1]

R. Alicandro and M. Ponsiglione, Ginzburg-Landau functionals and renormalized energy: A revised $\Gamma$-convergence approach, J. Funct. Anal., 266 (2014), 4890-4907. doi: 10.1016/j.jfa.2014.01.024.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser Verlag, Basel-Boston-Berlin, 2005.  Google Scholar

[3]

A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, 1994.  Google Scholar

[4]

F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices, 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]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1995. Google Scholar

[6]

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, 1999. Google Scholar

[7]

W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Physica D, 77 (1994), 383-404. doi: 10.1016/0167-2789(94)90298-4.  Google Scholar

[8]

W. E and P. Zhang, A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit, Methods and Applications of Analysis, 13 (2006), 181-198. doi: 10.4310/MAA.2006.v13.n2.a5.  Google Scholar

[9]

J. L. Ericksen, Conservation laws for liquid crystals, Journal of Rheology, 5 (1961), 23-34. doi: 10.1122/1.548883.  Google Scholar

[10]

I. Fatkullin and V. Slastikov, On spatial variations of nematic ordering, Physica D, 237 (2008), 2577-2586. doi: 10.1016/j.physd.2008.03.048.  Google Scholar

[11]

I. Fatkullin and V. Slastikov, Vortices in two-dimensional nematics, Communications in Mathematical Sciences, 7 (2009), 917-938. doi: 10.4310/CMS.2009.v7.n4.a6.  Google Scholar

[12]

F. M. Leslie, Some constitutive equations for liquid crystals, Archive for Rational Mechanics and Analysis, 28 (1968), 265-283. doi: 10.1007/BF00251810.  Google Scholar

[13]

F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices, 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]

J. C. Neu, Vortices in complex scalar fields, Physica D, 43 (1990), 385-406. doi: 10.1016/0167-2789(90)90143-D.  Google Scholar

[15]

E. Sandier, Lower bounds for the energy of unit vector fields and applications, Journal of Functional Analysis, 152 (1998), 379-403. doi: 10.1006/jfan.1997.3170.  Google Scholar

[16]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Communications on Pure and Applied Mathematics, 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.  Google Scholar

[17]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, AMS, 58, 1992. Google Scholar

[18]

P. Zhang W. Wang and Z. Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation, arXiv:1206.5480, 2013. Google Scholar

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