A gradient flow approach to an evolution problem arising in superconductivity

Luigi Ambrosio, Sylvia Serfaty

Research output: Contribution to journalArticle

Abstract

We study an evolution equation proposed by Chapman, Rubinstein, and Schatzman as a mean-field model for the evolution of the vortex density in a superconductor. We treat the case of a bounded domain where vortices can exit or enter the domain. We show that the equation can be derived rigorously as the gradient flow of some specific energy for the Riemannian structure induced by the Wasserstein distance on probability measures. This leads us to some existence and uniqueness results and energy-dissipation identities. We also exhibit some "entropies" that decrease through the flow and allow us to get regularity results (solutions starting in Lp, p > 1, remain in Lp).

Original languageEnglish (US)
Pages (from-to)1495-1539
Number of pages45
JournalCommunications on Pure and Applied Mathematics
Volume61
Issue number11
DOIs
StatePublished - Nov 2008

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Evolution Problems
Gradient Flow
Superconductivity
Vortex
Vortex flow
Wasserstein Distance
Mean-field Model
Existence and Uniqueness Results
Superconductor
Energy Dissipation
Superconducting materials
Probability Measure
Evolution Equation
Bounded Domain
Energy dissipation
Entropy
Regularity
Decrease
Energy

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

A gradient flow approach to an evolution problem arising in superconductivity. / Ambrosio, Luigi; Serfaty, Sylvia.

In: Communications on Pure and Applied Mathematics, Vol. 61, No. 11, 11.2008, p. 1495-1539.

Research output: Contribution to journalArticle

Ambrosio, Luigi ; Serfaty, Sylvia. / A gradient flow approach to an evolution problem arising in superconductivity. In: Communications on Pure and Applied Mathematics. 2008 ; Vol. 61, No. 11. pp. 1495-1539.
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