no. 2
Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 2, pp. 217-246

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.

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     author = {Ambrosio, Luigi and Mainini, Edoardo and Serfaty, Sylvia},
     title = {Gradient flow of the {Chapman{\textendash}Rubinstein{\textendash}Schatzman} model for signed vortices},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {217--246},
     year = {2011},
     publisher = {Elsevier},
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     doi = {10.1016/j.anihpc.2010.11.006},
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Ambrosio, Luigi; Mainini, Edoardo; Serfaty, Sylvia. Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 2, pp. 217-246. doi: 10.1016/j.anihpc.2010.11.006

[1] F.J. Almgren, J. Taylor, L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim. 31 no. 2 (1993), 387-438 | MR | Zbl

[2] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Clarendon Press, Oxford University Press, New York (2000) | MR | Zbl

[3] L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Spaces of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2005) | MR | Zbl

[4] L. Ambrosio, S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity, Comm. Pure Appl. Math. 61 no. 11 (2008), 1495-1539 | MR | Zbl

[5] F. Bethuel, D. Smets, A remark on the Cauchy problem for the 2D Gross–Pitaevskii equation with nonzero degree at infinity, Differential Integral Equations 20 no. 3 (2007), 325-338 | MR | Zbl

[6] J.S. Chapman, J. Rubinstein, M. Schatzman, A mean-field model for superconducting vortices, Eur. J. Appl. Math. 7 no. 2 (1996), 97-111 | MR | Zbl

[7] J.M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4 no. 3 (1991), 553-586 | MR

[8] R. Diperna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511-547 | MR | EuDML | Zbl

[9] Q. Du, P. Zhang, Existence of weak solutions to some vortex density models, SIAM J. Math. Anal. 34 (2003), 1279-1299 | MR | Zbl

[10] W. E, Dynamics of vortex-liquids in Ginzburg–Landau theories with applications to superconductivity, Phys. Rev. B 50 no. 3 (1994), 1126-1135 | MR

[11] R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal. 29 (1998), 1-17 | MR | Zbl

[12] F.H. Lin, P. Zhang, On the hydrodynamic limit of Ginzburg–Landau vortices, Discrete Cont. Dyn. Systems 6 (2000), 121-142 | MR | Zbl

[13] E. Mainini, A global uniqueness result for an evolution problem arising in superconductivity, Boll. Unione Mat. Ital. (9) II no. 2 (2009), 509-528 | MR | Zbl

[14] E. Mainini, A description of transport cost for signed measures, preprint, 2010. | MR

[15] R. Mccann, A convexity principle for interacting gases, Adv. Math. 128 (1997), 153-179 | MR | Zbl

[16] N. Masmoudi, P. Zhang, Global solutions to vortex density equations arising from sup-conductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 no. 4 (2005), 441-458 | MR | EuDML | Zbl | Numdam

[17] F. Otto, The geometry of dissipative evolution equations: the porous-medium equation, Comm. Partial Differential Equations 26 (2001), 101-174 | MR | Zbl

[18] Y. Ovchinnikov, I.M. Sigal, The energy of Ginzburg–Landau vortices, Eur. J. Appl. Math. 13 (2002), 153-178 | MR | Zbl

[19] E. Sandier, S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. Ec. Norm. Super. (4) 33 (2000), 561-592 | MR | EuDML | Zbl

[20] E. Sandier, S. Serfaty, Limiting vorticities for the Ginzburg–Landau equations, Duke Math. J. 117 (2003), 403-446 | MR | Zbl

[21] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics vol. 58, American Mathematical Society, Providence, RI (2003) | MR | Zbl

[22] C. Villani, Optimal Transport, Old and New, Springer-Verlag (2008) | MR | Zbl

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