Renormalized Energy Between Vortices in Some Ginzburg–Landau Models on 2-Dimensional Riemannian Manifolds

Ignat, Radu; Jerrard, Robert L.

doi:10.1007/s00205-020-01598-0

Cited by 28 publications

(54 citation statements)

References 37 publications

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“…A similar phenomenon arises for tangent vector fields on a closed manifold, due to the Poincaré-Hopf theorem (see e.g. [46]). Based on the analogy with the Ginzburg-Landau case (see e.g.…”

Section: Motivation: Variational Problems For Materials Sciencementioning

confidence: 59%

Topological singularities for vector-valued Sobolev maps and applications

Canevari¹,

Orlandi²

2021

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…A similar phenomenon arises for tangent vector fields on a closed manifold, due to the Poincaré-Hopf theorem (see e.g. [46]). Based on the analogy with the Ginzburg-Landau case (see e.g.…”

Section: Motivation: Variational Problems For Materials Sciencementioning

confidence: 59%

Topological singularities for vector-valued Sobolev maps and applications

Canevari¹,

Orlandi²

2021

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…A similar phenomenon arises for tangent vector fields on a closed manifold, due to the Poincaré-Hopf theorem (see e.g. [34]). The analysis of the Ginzburg-Landau case shows that the energy of minimisers (and other critical points) concentrates, to leading order, on a n-dimensional surface; see e.g.…”

Section: Introductionmentioning

confidence: 59%

Topological Singular Set of Vector-Valued Maps, II: $$\varGamma $$-convergence for Ginzburg–Landau type functionals

Canevari

Orlandi

2021

Arch Rational Mech Anal

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We prove a $$\varGamma $$ Γ -convergence result for a class of Ginzburg–Landau type functionals with $${\mathscr {N}}$$ N -well potentials, where $${\mathscr {N}}$$ N is a closed and $$(k-2)$$ ( k - 2 ) -connected submanifold of $${\mathbb {R}}^m$$ R m , in arbitrary dimension. This class includes, for instance, the Landau-de Gennes free energy for nematic liquid crystals. The energy density of minimisers, subject to Dirichlet boundary conditions, converges to a generalised surface (more precisely, a flat chain with coefficients in $$\pi _{k-1}({\mathscr {N}})$$ π k - 1 ( N ) ) which solves the Plateau problem in codimension k. The analysis relies crucially on the set of topological singularities, that is, the operator $${\mathbf {S}}$$ S we introduced in the companion paper [17].

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“…When χ(M ) = 0 the analysis is more involved as one has to deal with the emergence of vortices. In this case, Ignat & Jerrard obtained (see [7]) a Γ-convergence result for F extr ε . After a proper scaling, what emerges in the limit ε → 0 is a renormalized energy that takes into account the interactions between the vortices and between the vortices and the geometry of the surface, both intrinsic and extrinsic.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

“…A Nematic Shell is a thin film of nematic liquid crystal coating a rigid and curved substrate M , typically modeled as a two-dimensional closed and oriented surface (see, e.g., [9], [12], [17]). In this framework, the preferred direction of the molecular alignment is usually described by a unit-norm vector field tangent to M and the energy F extr ε is a relaxation (relaxing in particular the unit-norm constraint) of the elastic energy (see [11], [10], [15], [16]) (7) F extr 0 (u) :=…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…However, due to the incompatibility for vector fields on M between the Sobolev regularity and the unit norm constraint, when ε 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 extr ε .…”

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confidence: 80%

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

Canevari¹,

Segatti²

2021

Preprint

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We consider the gradient flow of a Ginzburg-Landau functional of the typewhich is defined for tangent vector fields (here D stands for the covariant derivative) on a closed surface M ⊆ R 3 and includes extrinsic effects via the shape operator S induced by the Euclidean embedding of M . The functional depends on the small parameter ε > 0. When ε is small it is clear from the structure of the Ginzburg-Landau functional that |u| 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 ε 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 extr ε . To this end we study the behavior when ε → 0 of the solutions of the (properly rescaled) gradient flow of F extr ε . In the limit ε → 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 ⊆ R 3 .

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Renormalized Energy Between Vortices in Some Ginzburg–Landau Models on 2-Dimensional Riemannian Manifolds

Cited by 28 publications

References 37 publications

Topological singularities for vector-valued Sobolev maps and applications

Topological singularities for vector-valued Sobolev maps and applications

Topological Singular Set of Vector-Valued Maps, II: $$\varGamma $$-convergence for Ginzburg–Landau type functionals

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

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