August  2022, 15(8): 2087-2116. doi: 10.3934/dcdss.2022116

Motion of vortices for the extrinsic Ginzburg-Landau flow for vector fields on surfaces

1. 

Dipartimento di Informatica, Università di Verona, Strada le Grazie 15, 37134 Verona, Italy

2. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy

* Corresponding author: Antonio Segatti

Dedicated to Maurizio Grasselli on the occasion of his 60th anniversary, with friendship and admiration

Received  December 2021 Revised  April 2022 Published  August 2022 Early access  May 2022

We consider the gradient flow of a Ginzburg-Landau functional of the type
$ F_ \varepsilon^{ \mathrm{extr}}(u): = \frac{1}{2}\int_M \left| {D u} \right|_g^2 + \left| { \mathscr{S} u} \right|^2_g +\frac{1}{2 \varepsilon^2}\left(\left| {u} \right|^2_g-1\right)^2 \mathrm{vol}_g $
which is defined for tangent vector fields (here
$ D $
stands for the covariant derivative) on a closed surface
$ M\subseteq \mathbb{R}^3 $
and includes extrinsic effects via the shape operator
$ \mathscr{S} $
induced by the Euclidean embedding of
$ M $
. The functional depends on the small parameter
$ \varepsilon>0 $
. When
$ \varepsilon $
is small it is clear from the structure of the Ginzburg-Landau functional that
$ \left| {u} \right|_g $
"prefers" to be close to
$ 1 $
. However, due to the incompatibility for vector fields on
$ M $
between the Sobolev regularity and the unit norm constraint, when
$ \varepsilon $
is close to
$ 0 $
, it is expected that a finite number of singular points (called vortices) having non-zero index emerges (when the Euler characteristic is non-zero). This intuitive picture has been made precise in the recent work by R. Ignat & R. Jerrard [7]. In this paper we are interested the dynamics of vortices generated by
$ F_ \varepsilon^{ \mathrm{extr}} $
. To this end we study the behavior when
$ \varepsilon\to 0 $
of the solutions of the (properly rescaled) gradient flow of
$ F_ \varepsilon^{ \mathrm{extr}} $
. In the limit
$ \varepsilon\to 0 $
we obtain the effective dynamics of the vortices. The dynamics, as expected, is influenced by both the intrinsic and extrinsic properties of the surface
$ M\subseteq \mathbb{R}^3 $
.
Citation: Giacomo Canevari, Antonio Segatti. Motion of vortices for the extrinsic Ginzburg-Landau flow for vector fields on surfaces. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2087-2116. doi: 10.3934/dcdss.2022116
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[2]

F. Bethuel, H. Brezis and F. Hélein, 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.

[3]

G. Canevari and A. Segatti, Dynamics of Ginzburg-Landau vortices for vector fields on surfaces, 2021, preprint. https://arXiv.org/abs/2108.01321.

[4]

G. Canevari and A. Segatti, Defects in nematic shells: A $\Gamma$-convergence discrete-to-continuum approach, Arch. Ration. Mech. Anal., 229 (2018), 125-186.  doi: 10.1007/s00205-017-1215-z.

[5]

G. CanevariA. Segatti and M. Veneroni, Morse's index formula in VMO for compact manifolds with boundary, J. Funct. Anal., 269 (2015), 3043-3082.  doi: 10.1016/j.jfa.2015.09.005.

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2001, Reprint of the 1998 edition.

[7]

R. Ignat and R. L. Jerrard, Renormalized energy between vortices in some Ginzburg-Landau models on 2-dimensional Riemannian manifolds, Arch. Ration. Mech. Anal., 239 (2021), 1577-1666.  doi: 10.1007/s00205-020-01598-0.

[8]

F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), 323-359.  doi: 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E.

[9]

T. C. Lubensky and J. Prost, Orientational order and vesicle shape, J. Phys. II France, 2 (1992), 371-382.  doi: 10.1051/jp2:1992133.

[10]

G. Napoli and L. Vergori, Extrinsic curvature effects on nematic shells, Phys. Rev. Lett., 108 (2012), 207803.  doi: 10.1103/PhysRevLett.108.207803.

[11]

G. Napoli and L. Vergori, Surface free energies for nematic shells, Phys. Rev. E, 85 (2012), 061701.  doi: 10.1103/PhysRevE.85.061701.

[12]

D. R. Nelson, Toward a tetravalent chemistry of colloids, Nano Lett., 2 (2002), 1125-1129.  doi: 10.1021/nl0202096.

[13]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.  doi: 10.1002/cpa.20046.

[14]

C. Scott, $L^p$ theory of differential forms on manifolds, Trans. Amer. Math. Soc., 347 (1995), 2075-2096.  doi: 10.2307/2154923.

[15]

A. SegattiM. Snarski and M. Veneroni, Equilibrium configurations of nematic liquid crystals on a torus, Phys. Rev. E, 90 (2014), 012501.  doi: 10.1103/PhysRevE.90.012501.

[16]

A. SegattiM. Snarski and M. Veneroni, Analysis of a variational model for nematic shells, Math. Models Methods Appl. Sci., 26 (2016), 1865-1918.  doi: 10.1142/S0218202516500470.

[17]

J. P. Straley, Liquid crystals in two dimensions, Phys. Rev. A, 4 (1971), 675-681.  doi: 10.1103/PhysRevA.4.675.

[18]

D. H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim., 15 (1977), 859-903.  doi: 10.1137/0315056.

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.

[2]

F. Bethuel, H. Brezis and F. Hélein, 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.

[3]

G. Canevari and A. Segatti, Dynamics of Ginzburg-Landau vortices for vector fields on surfaces, 2021, preprint. https://arXiv.org/abs/2108.01321.

[4]

G. Canevari and A. Segatti, Defects in nematic shells: A $\Gamma$-convergence discrete-to-continuum approach, Arch. Ration. Mech. Anal., 229 (2018), 125-186.  doi: 10.1007/s00205-017-1215-z.

[5]

G. CanevariA. Segatti and M. Veneroni, Morse's index formula in VMO for compact manifolds with boundary, J. Funct. Anal., 269 (2015), 3043-3082.  doi: 10.1016/j.jfa.2015.09.005.

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2001, Reprint of the 1998 edition.

[7]

R. Ignat and R. L. Jerrard, Renormalized energy between vortices in some Ginzburg-Landau models on 2-dimensional Riemannian manifolds, Arch. Ration. Mech. Anal., 239 (2021), 1577-1666.  doi: 10.1007/s00205-020-01598-0.

[8]

F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), 323-359.  doi: 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E.

[9]

T. C. Lubensky and J. Prost, Orientational order and vesicle shape, J. Phys. II France, 2 (1992), 371-382.  doi: 10.1051/jp2:1992133.

[10]

G. Napoli and L. Vergori, Extrinsic curvature effects on nematic shells, Phys. Rev. Lett., 108 (2012), 207803.  doi: 10.1103/PhysRevLett.108.207803.

[11]

G. Napoli and L. Vergori, Surface free energies for nematic shells, Phys. Rev. E, 85 (2012), 061701.  doi: 10.1103/PhysRevE.85.061701.

[12]

D. R. Nelson, Toward a tetravalent chemistry of colloids, Nano Lett., 2 (2002), 1125-1129.  doi: 10.1021/nl0202096.

[13]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.  doi: 10.1002/cpa.20046.

[14]

C. Scott, $L^p$ theory of differential forms on manifolds, Trans. Amer. Math. Soc., 347 (1995), 2075-2096.  doi: 10.2307/2154923.

[15]

A. SegattiM. Snarski and M. Veneroni, Equilibrium configurations of nematic liquid crystals on a torus, Phys. Rev. E, 90 (2014), 012501.  doi: 10.1103/PhysRevE.90.012501.

[16]

A. SegattiM. Snarski and M. Veneroni, Analysis of a variational model for nematic shells, Math. Models Methods Appl. Sci., 26 (2016), 1865-1918.  doi: 10.1142/S0218202516500470.

[17]

J. P. Straley, Liquid crystals in two dimensions, Phys. Rev. A, 4 (1971), 675-681.  doi: 10.1103/PhysRevA.4.675.

[18]

D. H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim., 15 (1977), 859-903.  doi: 10.1137/0315056.

[1]

Leonid Berlyand, Volodymyr Rybalko, Nung Kwan Yip. Renormalized Ginzburg-Landau energy and location of near boundary vortices. Networks and Heterogeneous Media, 2012, 7 (1) : 179-196. doi: 10.3934/nhm.2012.7.179

[2]

Hassen Aydi, Ayman Kachmar. Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II. Communications on Pure and Applied Analysis, 2009, 8 (3) : 977-998. doi: 10.3934/cpaa.2009.8.977

[3]

Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075

[4]

Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229

[5]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205

[6]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks and Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715

[7]

Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121

[8]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345

[9]

Leonid Berlyand, Petru Mironescu. Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Networks and Heterogeneous Media, 2008, 3 (3) : 461-487. doi: 10.3934/nhm.2008.3.461

[10]

N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647

[11]

Leonid Berlyand, Volodymyr Rybalko. Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes. Networks and Heterogeneous Media, 2013, 8 (1) : 115-130. doi: 10.3934/nhm.2013.8.115

[12]

Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 835-851. doi: 10.3934/dcdss.2020048

[13]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713

[14]

Satoshi Kosugi, Yoshihisa Morita. Phase pattern in a Ginzburg-Landau model with a discontinuous coefficient in a ring. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 149-168. doi: 10.3934/dcds.2006.14.149

[15]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871

[16]

Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173

[17]

Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280

[18]

N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476

[19]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure and Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665

[20]

Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754

2021 Impact Factor: 1.865

Article outline

[Back to Top]