Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices

Abstract

We continue the study of Ambrosio and Serfaty (2008) [4] on the Chapman-Rubinstein-Schatzman-E evolution model for superconductivity, viewed as a gradient flow on the space of measures equipped with the quadratic Wasserstein structure. In Ambrosio and Serfaty (2008) [4] we considered the case of positive (probability) measures, while here we consider general real measures, as in the physical model. Understanding the evolution as a gradient flow in this context gives rise to several new questions, in particular how to define a "Wasserstein" distance for signed measures. We generalize the minimizing movement scheme of Ambrosio et al. (2005) [3] in this context, we show the entropy argument of Ambrosio and Serfaty (2008) [4] still carries through, and derive an evolution equation for the measure which contains an error term compared to the Chapman-Rubinstein-Schatzman-E model. Moreover, we also show the same applies to a very similar dissipative model on the whole plane.

Vortex Liquids and the Ginzburg–landau Equation

“…Ambrosio and Serfaty [1] and Ambrosio, Mainini, and Serfaty [2] study them as a metric gradient flow in the space of measures with the Wasserstein distance as the natural metric; however, they do not obtain the convergence. Even when it becomes possible to carry out the program outlined in the survey of Serfaty [47] and to directly obtain the Wasserstein gradient flow studied in [1, 2] from the Gorkov-Eliashberg equation by the Γ -convergence of a gradient flow type result, we believe that our approach will still be useful.…”

“…Our approach of using differential identities and explicit estimates follows the program of the second author and Jerrard [23] for the Gross-Pitaevsky equation i∂ t u = ∆u + 1 ε 2 u 1 − |u| 2 . Surprisingly, implementing this approach for (1) is more challenging and requires several new estimates.…”

“…Let u : [0, ∞) × Ω → C satisfy the scaled Ginzburg-Landau equation 1 |log ε| ∂ t u = ∆u + 1 ε 2 u 1 − |u| 2 (1) with either Dirichlet boundary conditions u = e inθ +iϕ on ∂Ω…”

Nonlinear Diffusion with Fractional Laplacian Operators

“…The authors prove existence and uniqueness of positive L ∞ solutions (they also prove existence of positivemeasure valued solutions). Existence with positive initial data of finite energy is also proven (for a slightly different model) by a gradient flow approach, in bounded domains of the plane by Ambrosio et al in [3] and in R 2 in [2].…”

Well-posedness for a mean field model of Ginzburg–Landau vortices with opposite degrees

“…4 we will very briefly recall the main elements of the theory. In [2,17], as well as in this paper, we continue the analysis in this framework. For positive solutions in the whole plane, the model reduces to a single equation of (1.1) (this happens when the vortex degree is uniform, say equal to 1, and also one recovers the model of E in this case).…”