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

Giacomo Canevari; Antonio Segatti

doi:10.3934/dcdss.2022116

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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

Giacomo Canevari ^1, and Antonio Segatti ^2,,

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 $

Keywords: Ginzburg-Landau, vector fields on surfaces, gradient flow of the renormalized energy, $ \Gamma $-convergence.

Mathematics Subject Classification: 35Q56, 37E35, 49J45, 58E20.

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