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

top
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.

How to cite

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.