Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature

Béthuel, Fabrice; Orlandi, Giandomenico; Smets, Didier

doi:10.4007/annals.2006.163.37

Cited by 67 publications

(15 citation statements)

References 41 publications

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“…The advantage of these PDE methods over the one we present here is that they also work for dispersive equations. In dimension 3, the limiting heat flow of Ginzburg-Landau is again the mean curvature (Brakke) flow; see [3,5,24].…”

Section: Introductionmentioning

confidence: 99%

Gamma‐convergence of gradient flows with applications to Ginzburg‐Landau

Sandier

Serfaty

2004

Comm Pure Appl Math

285

387

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We present a method to prove convergence of gradient flows of families of energies that -converge to a limiting energy. It provides lower-bound criteria to obtain the convergence that correspond to a sort of C 1 -order -convergence of functionals. We then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the Ginzburg-Landau energy in dimension 2, retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field.

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Section: Introductionmentioning

confidence: 99%

Gamma‐convergence of gradient flows with applications to Ginzburg‐Landau

Sandier

Serfaty

2004

Comm Pure Appl Math

285

387

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“…A large number of papers have been devoted to this type of results, using several methods. Without any claim for completeness, we mention [1, 3, 8-15, 23-25, 31, 38, 42, 43]; observe that also similar questions have been addressed also for the stationary equation (see, e.g., [29,36,39,40]) and for systems (see, e.g., [6,33]). For a comprehensive account of literature on this subject, also containing the description of main results obtained and various methods used, we refer the reader to [43] and references therein.…”

Section: A Typical Example Ismentioning

confidence: 99%

Allen-Cahn approximation of mean curvature flow in Riemannian manifolds I, uniform estimates

Pisante¹,

Punzo²

2016

Annali Scuola Normale Superiore - Classe Di Scienze

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We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial conditions we show nonpositivity of the limiting energy discrepancy. This in turn allows us to prove an almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) which gives a local uniform control of the energy densities at small scales.These results will be used in [41] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in Riemannian manifolds.

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“…Many authors studied the dynamics properties of Ginzburg–Landau vortices under some general assumption on u ( x ,0) (cf. , , and ).…”

Section: Introductionmentioning

confidence: 99%

On an initial‐boundary value problem for the p‐Ginzburg–Landau system

Lei

2014

Math Methods in App Sciences

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This paper is concerned with the asymptotic behavior of the decreasing energy solution uε to a p‐Ginzburg–Landau system with the initial‐boundary data for p > 4/3. It is proved that the zeros of uε in the parabolic domain G × (0,T] are located near finite lines {ai}×(0,T]. In particular, all the zeros converge to these lines when the parameter ε goes to zero. In addition, the author also considers the uniform energy estimation on a domain far away from the zeros. At last, the Hölder convergence of uε to a heat flow of p‐harmonic map on this domain is proved when p > 2. Copyright © 2014 John Wiley & Sons, Ltd.

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Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature

Cited by 67 publications

References 41 publications

Gamma‐convergence of gradient flows with applications to Ginzburg‐Landau

Gamma‐convergence of gradient flows with applications to Ginzburg‐Landau

Allen-Cahn approximation of mean curvature flow in Riemannian manifolds I, uniform estimates

On an initial‐boundary value problem for the p‐Ginzburg–Landau system

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