# American Institute of Mathematical Sciences

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

## Diffusive transport in two-dimensional nematics

 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.
Citation: Ibrahim Fatkullin, Valeriy Slastikov. Diffusive transport in two-dimensional nematics. Discrete and 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. [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. [3] A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, 1994. [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. [5] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1995. [6] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, 1999. [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. [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. [9] J. L. Ericksen, Conservation laws for liquid crystals, Journal of Rheology, 5 (1961), 23-34. doi: 10.1122/1.548883. [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. [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. [12] F. M. Leslie, Some constitutive equations for liquid crystals, Archive for Rational Mechanics and Analysis, 28 (1968), 265-283. doi: 10.1007/BF00251810. [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. [14] J. C. Neu, Vortices in complex scalar fields, Physica D, 43 (1990), 385-406. doi: 10.1016/0167-2789(90)90143-D. [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. [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. [17] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, AMS, 58, 1992. [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.

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##### 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. [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. [3] A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, 1994. [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. [5] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1995. [6] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, 1999. [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. [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. [9] J. L. Ericksen, Conservation laws for liquid crystals, Journal of Rheology, 5 (1961), 23-34. doi: 10.1122/1.548883. [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. [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. [12] F. M. Leslie, Some constitutive equations for liquid crystals, Archive for Rational Mechanics and Analysis, 28 (1968), 265-283. doi: 10.1007/BF00251810. [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. [14] J. C. Neu, Vortices in complex scalar fields, Physica D, 43 (1990), 385-406. doi: 10.1016/0167-2789(90)90143-D. [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. [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. [17] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, AMS, 58, 1992. [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.
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