Well-posedness for mean-field evolutions arising in superconductivity

  • Julian Fischer

    Institute of Science and Technology Austria (IST Austria), Klosterneuburg, Austria
  • Mitia Duerinckx

    Université Libre de Bruxelles (ULB), Brussels, Belgium; Laboratoire Jacques-Louis-Lions, Université Pierre et Marie Curie (UPMC), Paris, France


We establish the existence of a global solution for a new family of fluid-like equations, which are obtained in certain regimes in [24] as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. We also consider general vortex-sheet initial data, and investigate the uniqueness and regularity properties of the solution. For some choice of parameters, the equation under investigation coincides with the so-called lake equation from 2D shallow water fluid dynamics, and our analysis then leads to a new existence result for rough initial data.

Cite this article

Julian Fischer, Mitia Duerinckx, Well-posedness for mean-field evolutions arising in superconductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 5, pp. 1267–1319

DOI 10.1016/J.ANIHPC.2017.11.004