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1. | Department of Mathematics, University of Arizona, Tucson, AZ 85721, United States |
2. | Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom |
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. |
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. |
[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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