Skip to main content
SpringerLink
Log in

A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a “mean field” equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the “elementary vortex patch”.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Wasserstein Space of Probability Measures. Birkäuser, Basel (2005)

    Google Scholar 

  2. Ambrosio, L., Serfaty, S.: A gradient flow approach to an evolution problem arising in superconductivity. Commun. Pure Appl. Math. 61(11), 1495–1539 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ambrosio, L., Mainini, E., Serfaty, S.: Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices. Ann. IHP Anal. non linéaire 28(2), 217–246 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Aubin, J.P.: Un théorème de compacité. C. R. Acad. Sci. 256, 5042–5044 (1963)

    MATH  MathSciNet  Google Scholar 

  5. Bénilan, P., Crandall, M.G.: Regularizing effects of homogeneous evolution equations. Contributions to Analysis and Geometry (suppl. to American Journal of Mathematical), pp. 23–39. Johns Hopkins University Press, Baltimore (1981)

  6. Bertozzi, A., Carrillo, J.A., Laurent, T.: Blow-up in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22, 683–710 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bertozzi, A., Garnett, J.B., Laurent, T.: Characterization of radially symmetric finite time blowup in multidimensional aggregation equations. SIAM J. Math. Anal. 44(2), 651–681 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bertozzi, A., Laurent, T., Léger, F.: Aggregation via Newtonian potential and aggregation patches. M3AS 22(Supp. 1), (2012, special issue)

  9. Bethuel, F., Smets, D.: A remark on the Cauchy Problem for the 2D Gross–Pitaevskii equation with non zero degree at infinity. Differ. Integral Equ. 20(3), 325–338 (2007)

    MATH  MathSciNet  Google Scholar 

  10. Biler, P., Imbert, C., Karch, G.: Barenblatt profiles for a nonlocal porous medium equation. C. R. Acad. Sci. Paris Ser. I(349), 641–645 (2011)

    Article  MathSciNet  Google Scholar 

  11. Biler, P., Karch, G., Monneau, R.: Nonlinear diffusion of dislocation density and self-similar solutions. Commun. Math. Phys. 294(1), 145–168 (2010). MR2575479

    Google Scholar 

  12. Blanchet, A., Dolbeault, J., Perthame, B.: Two dimensional Keller–Segel model in \(R^2\): optimal critical mass and qualitative properties of the solution. Electron. J. Differ. Equ. 44, 1–33 (2006)

    MathSciNet  Google Scholar 

  13. Caffarelli, L.A., Soria, F., Vazquez, J.L.: Regularity of solutions of the fractional porous medium flow. arXiv 1201.6048v1 (2012)

  14. Caffarelli, L.A., Vazquez, J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Rational Mech. Anal. 202, 537–565 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Caffarelli, L.A., Vazquez, J.L.: Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Cont. Dyn. Syst. A 29(4), 1393–1404 (2011)

    MATH  MathSciNet  Google Scholar 

  16. Chapman, S.J., Rubinstein, J., Schatzman, M.: A mean-field model for superconducting vortices. Eur. J. Appl. Math. 7(2), 97–111 (1996)

    Google Scholar 

  17. Weinan, E.: Dynamics of vortex-liquids in Ginzburg-Landau theories with applications to superconductivity. Phys. Rev. B 50(3), 1126–1135 (1994)

    Google Scholar 

  18. Elliott, C.M., Herrero, M.A., King, J.R., Ockendon, J.R.: The mesa problem: diffusion patterns for \(u_t=\nabla \cdot (u^m\nabla u)\) as \(m\rightarrow +\infty \). IMA J. Appl. Math. 37(2), 147–154 (1986)

  19. Friedman, A., Höllig, K.: On the mesa problem. J. Math. Anal. Appl. 123(2), 564–571 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  20. Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interaction I. Macroscopic limits. J. Stat. Phys. 87, 37–61 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interaction II. Interface motion. SIAM J. Appl. Math. 58, 170729 (1998)

    Article  MathSciNet  Google Scholar 

  22. Giacomin, G., Lebowitz, J., Marra, R.: Macroscopic evolution of particle systems with short and long-range interactions. Nonlinearity 13(6), 2143–2162 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Head, A.K.: Dislocation group dynamics II. Similarity solutions of the continuum approximation. Philos. Mag. 26, 65–72 (1972)

    Article  Google Scholar 

  24. Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992)

    MATH  Google Scholar 

  25. Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)

    Article  MATH  Google Scholar 

  26. Kruzhkov, S.N.: First order quasilinear equations with several independent variables. (Russian) Mat. Sb. (N.S.) 81(123), 228–255 (1970)

  27. Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasilinear equations of parabolic type. In: Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)

  28. Landkof, N.S.: Foundations of Modern Potential Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 180. Translated from the Russian by A. P. Doohovskoy. Springer, New York (1972)

  29. Lax, P.D.: Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  30. Lieberman, G.M.: Second order parabolic differential equations. World Scientific, River Edge (1996)

    Book  MATH  Google Scholar 

  31. Lin, F.H., Zhang, P.: On the hydrodynamic limit of Ginzburg–Landau vortices. Discrete Cont. Dyn. Syst. 6, 121–142 (2000)

    MATH  Google Scholar 

  32. Liu, T.-P., Pierre, M.: Source-solutions and asymptotic behavior in conservation laws. J. Differ. Equ. 51(3), 419–441 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  33. Loeper, G.: Uniqueness of the solution to the Vlasov–Poisson system with bounded density. J. Math. Pures Appl. (9) 86(1), 68–79 (2006)

    Google Scholar 

  34. Majda, A., Bertozzi, A.: Vorticity and incompressible flow. In: Cambridge Texts in Applied Mathematics. (2002)

  35. Masmoudi, N., Zhang, P.: Global solutions to vortex density equations arising from sup-conductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(4), 441–458 (2005)

    Google Scholar 

  36. Mazya, V., Shaposhnikova, T.: On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195(2), 230–238 (2002)

    Article  MathSciNet  Google Scholar 

  37. Ovchinnikov, Y., Sigal, I.M.: The energy of Ginzburg–Landau vortices. Eur. J. Appl. Math. 13, 153–178 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  38. Schätzle, R., Styles, V.: Analysis of a mean field model of superconducting vortices. Eur. J. Appl. Math. 10(4), 319–352 (1999)

    Article  MATH  Google Scholar 

  39. Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. (4) 146, 65–96 (1987)

    Google Scholar 

  40. Stein, E.M.: Singular integrals and differentiability properties of functions. In: Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, NJ (1970)

  41. Vázquez, J.L.: The porous medium equation. In: Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007)

  42. Vázquez, J.L.: Nonlinear diffusion with fractional Laplacian operators. In: Holden, H., Karlsen, K.H. (eds.) Nonlinear Partial Differential Equations: the Abel Symposium 2010, pp. 271–298. Springer, Berlin (2012)

Download references

Acknowledgments

We would like to thank J. A. Carrillo for pointing out to us the reference [8] after a first draft of this paper was written. S. S. thanks the Universidad Autónoma de Madrid for its hospitality that allowed this work to be completed. S. S. was supported by a EURYI award, and JLV by Spanish Project MTM2008-06326-C02-01.

Author information

Authors and Affiliations

  1. UPMC Univ Paris 06, CNRS, UMR 7598 Laboratoire Jacques-Louis Lions, 75005 , Paris, France

    Sylvia Serfaty

  2. Courant Institute, New York University, 251 Mercer st, New York, NY, 10012, USA

    Sylvia Serfaty

  3. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 , Madrid, Spain

    Juan Luis Vázquez

Authors
  1. Sylvia Serfaty
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Luis Vázquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sylvia Serfaty.

Additional information

Communicated by L. Ambrosio.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Serfaty, S., Vázquez, J.L. A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators. Calc. Var. 49, 1091–1120 (2014). https://doi.org/10.1007/s00526-013-0613-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-013-0613-9

Mathematics Subject Classification (2000)

Search

Navigation