A Global Uniqueness Result for an Evolution Problem Arising in Superconductivity

Edoardo Mainini

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 2, page 509-528
  • ISSN: 0392-4041

Abstract

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We consider an energy functional on measures in ℝ 2 arising in superconductivity as a limit case of the well-known Ginzburg Landau functionals. We study its gradient flow with respect to the Wasserstein metric of probability measures, whose corresponding time evolutive problem can be seen as a mean field model for the evolution of vortex densities. Improving the analysis made in [AS], we obtain a new existence and uniqueness result for the evolution problem.

How to cite

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Mainini, Edoardo. "A Global Uniqueness Result for an Evolution Problem Arising in Superconductivity." Bollettino dell'Unione Matematica Italiana 2.2 (2009): 509-528. <http://eudml.org/doc/290562>.

@article{Mainini2009,
abstract = {We consider an energy functional on measures in $\mathbb\{R\}^\{2\}$ arising in superconductivity as a limit case of the well-known Ginzburg Landau functionals. We study its gradient flow with respect to the Wasserstein metric of probability measures, whose corresponding time evolutive problem can be seen as a mean field model for the evolution of vortex densities. Improving the analysis made in [AS], we obtain a new existence and uniqueness result for the evolution problem.},
author = {Mainini, Edoardo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {509-528},
publisher = {Unione Matematica Italiana},
title = {A Global Uniqueness Result for an Evolution Problem Arising in Superconductivity},
url = {http://eudml.org/doc/290562},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Mainini, Edoardo
TI - A Global Uniqueness Result for an Evolution Problem Arising in Superconductivity
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/6//
PB - Unione Matematica Italiana
VL - 2
IS - 2
SP - 509
EP - 528
AB - We consider an energy functional on measures in $\mathbb{R}^{2}$ arising in superconductivity as a limit case of the well-known Ginzburg Landau functionals. We study its gradient flow with respect to the Wasserstein metric of probability measures, whose corresponding time evolutive problem can be seen as a mean field model for the evolution of vortex densities. Improving the analysis made in [AS], we obtain a new existence and uniqueness result for the evolution problem.
LA - eng
UR - http://eudml.org/doc/290562
ER -

References

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  1. AMBROSIO, L. - GANGBO, W., Hamiltonian ODE's in the Wasserstein space of probability measures, Comm. Pure Appl. Math., LXI, no. 1 (2008), 18-53. Zbl1132.37028MR2361303DOI10.1002/cpa.20188
  2. AMBROSIO, L. - SERFATY, S., A gradient flow approach to an evolution problem arising in superconductivity, Comm. Pure Appl. Math., LXI, no. 11 (2008), 1495-1539. Zbl1171.35005MR2444374DOI10.1002/cpa.20223
  3. AMBROSIO, L. - GIGLI, N. - SAVARÉ, G., Gradient flows in metric spaces and in the spaces of probability measures, Lectures in Mathematics ETH Zu Èrich, Birkhäuser Verlag, Basel (2005). MR2129498
  4. CHAPMAN, J. S. - RUBINSTEIN, J. - SCHATZMAN, M., A mean-field model for superconducting vortices, Eur. J. Appl. Math., 7, no. 2 (1996), 97-111. Zbl0849.35135MR1388106DOI10.1017/S0956792500002242
  5. JORDAN, R. - KINDERLEHRER, D. - OTTO, F., The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. Zbl0915.35120MR1617171DOI10.1137/S0036141096303359
  6. OTTO, F., The geometry of dissipative evolution equations: the porous-medium equation, Comm. PDE, 26 (2001), 101-174. Zbl0984.35089MR1842429DOI10.1081/PDE-100002243
  7. SANDIER, E. - SERFATY, S., A Rigorous Derivation of a Free-Boundary Problem Arising in Superconductivity, Ann. Scien. Ecole Normale Supérieure, 4e ser 33 (2000), 561-592. Zbl1174.35552MR1832824DOI10.1016/S0012-9593(00)00122-1
  8. SANDIER, E. - SERFATY, S., Limiting Vorticities for the Ginzburg-Landau equations, Duke Math. J., 117 (2003), 403-446. Zbl1035.82045MR1979050DOI10.1215/S0012-7094-03-11732-9
  9. VILLANI, C., Topics in optimal transportation, Graduate Studies in Mathematics58, American Mathematical Society, Providence, RI, (2003). Zbl1106.90001MR1964483DOI10.1007/b12016
  10. YUDOVICH, V., Nonstationary flow of an ideal incompressible liquid, Zhurn. Vych. Mat., 3 (1963), 1032-1066. MR158189
  11. YUDOVICH, V., Some bounds for solutions of elliptic equations, Mat. Sb., 59 (1962), 229-244; English transl. in Amer. Mat. Soc. Transl. (2), 56 (1962). MR149062

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