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

Sandier, Étienne; Serfaty, Sylvia

doi:10.1002/cpa.20046

Cited by 285 publications

(398 citation statements)

References 36 publications

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“…This result provides a purely quantitative approach to the kinetic energy lower bounds that are found in [30,18,41], each of which relies on compactness properties to get a lower bound. To establish this result we make quantitative the kinetic energy estimate of [30], who used the differential identity for the energy density, along with a limiting result on the equipartitioning of potential energy.…”

Section: Results In the Following We Letmentioning

confidence: 99%

“…Although (1) provides a fertile ground to test the mathematics of the Gorkov-Eliashberg equations, the more physical problem entails looking at the hydrodynamic limit of (4). For the Gorkov-Eliashberg equations (4), corresponding proofs of the vortex motion law are due to the second author [49] for O(1) fields and Sandier and Serfaty [41] for larger fields, following the formal asymptotic work of [38]. Formally, it was shown by Chapman, Rubinstein, and Schatzman [7] that the hydrodynamic limit of the associated ODE arising from the vortex motion law of (4) converges to a weak solution of…”

Section: Results In the Following We Letmentioning

confidence: 99%

“…This structure was exploited to give a more abstract proof of the motion law by Sandier and Serfaty [41] in their Γ -convergence of a gradient flow framework. In recent years there have been significant advances in understanding the dynamics of a finite numbers of vortices by Bethuel, Orlandi, and Smets [4] on R 2 and by Serfaty [46] on bounded domains.…”

Section: Introductionmentioning

confidence: 99%

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Vortex Liquids and the Ginzburg–landau Equation

Kurzke

Spirn

2014

Forum math. Sigma

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We establish vortex dynamics for the time-dependent Ginzburg-Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.

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Section: Results In the Following We Letmentioning

confidence: 99%

Section: Results In the Following We Letmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Vortex Liquids and the Ginzburg–landau Equation

Kurzke

Spirn

2014

Forum math. Sigma

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“…Variational-evolution structure, such as in the case of gradient flows and variational rate-independent systems, also facilitates limits [28,51,53,54,67,70,71].…”

Section: Introductionmentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with nondissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the smallnoise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. Mathematics Subject Classification

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“…[4,28,30,39,38,40]). The inherent convexity and lower-semicontinuity properties of this type of formulation provide handles for such limit passages that are similar to the well-known results for elliptic systems-as we show below.…”

Section: Introductionmentioning

confidence: 99%

Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

et al. 2011

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We study a singular-limit problem arising in the modelling of chemical reactions. At finite ε > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/ε, and in the limit ε → 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier et al. (SIAM J Math Anal, 42(4):1805-1825, 2010, using the linear structure of the equation. In this study we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the ε-problem converge to a solution of the limiting problem.

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

Cited by 285 publications

References 36 publications

Vortex Liquids and the Ginzburg–landau Equation

Vortex Liquids and the Ginzburg–landau Equation

Variational approach to coarse-graining of generalized gradient flows

Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

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