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mSQG equations in distributional spaces and point vortex approximation

  • Franco FlandoliEmail author
  • Martin Saal
Article
  • 3 Downloads

Abstract

Existence of distributional solutions of a modified surface quasi-geostrophic equation is proved for \(\mu \)-almost every initial condition, where \(\mu \) is a suitable Gaussian measure. The result is the by-product of existence of a stationary solution with white noise marginal. This solution is constructed as a limit of random point vortices, uniformly distributed and rescaled according to the Central Limit Theorem.

Mathematics Subject Classification

Primary 60H15 Secondary 35Q86 35R60 76B03 

Notes

References

  1. 1.
    S. Albeverio, A. B. Cruzeiro, Global flows with invariant (Gibbs) measures for Euler and Navier–Stokes two-dimensional fluids, Comm. Math. Phys. 129 (1990), 431–444,  https://doi.org/10.1007/BF02097100.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    T. Buckmaster, A. Nahmod, G. Staffilani, K. Widmayer, The Surface Quasi-geostrophic Equation With Random Diffusion, to appear in International Mathematics Research Notices (2018),  https://doi.org/10.1093/imrn/rny261.
  3. 3.
    T. Buckmaster, S. Shkoller, V. Vicol, Nonuniqueness of weak solutions to the SQG equation, arXiv:1610.00676 (2016).
  4. 4.
    G. Cavallaro, R. Garra, C. Marchioro, Localization and stability of active scalar flows, Riv. Mat. Univ. Parma 4 (2013), no. 1, 175–196.MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Castro, D. Córdoba, J. Gómez-Serrano, Global smooth solutions for the inviscid SQG equation, arXiv:1603.03325v3 (2017).
  6. 6.
    D. Chae, P. Constantin, D. Córdoba, F. Gancedo, J. Wu, Generalized surface quasi-geostrophic equations with singular velocities, Comm. Pure Appl. Math. 65 (2012), no. 8, 1037–1066,  https://doi.org/10.1002/cpa.21390.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D. Chae, P. Constantin, J. Wu, Inviscid Models Generalizing the Two-dimensional Euler and the Surface Quasi-geostrophic Equations, Riv. Math. Univ. Parma 202 (2011), 35–62,  https://doi.org/10.1007/s00205-011-0411-5.MathSciNetzbMATHGoogle Scholar
  8. 8.
    P. Constantin, Analysis of Hydrodynamic Models CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM-Society for Industrial and Applied Mathematics, 2017,  https://doi.org/10.1137/1.9781611974805.Google Scholar
  9. 9.
    P. Constantin, A. Majda, E. G. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), 1495–1533,  https://doi.org/10.1088/0951-7715/7/6/001.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    P. Constantin, A.J. Majda, E. G. Tabak, Singular front formation in a model for quasigeostrophic flow, Phys. Fluids 6 (1994), no. 1, 9–11,  https://doi.org/10.1063/1.868050.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    P. Constantin, H. Nguyen, Global Weak Solutions for SQG in Bounded Domains, Comm. Pure Appl. Math. 71 (2018), 2323–2333,  https://doi.org/10.1002/cpa.21720.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Córdoba, D. Córdoba, Uniqueness for SQG patch solutions, Trans. Amer. Math. Soc. Ser. B 5 (2018), 1–31,  https://doi.org/10.1090/btran/20.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    D. Córdoba, M. Fontelos, A. Mancho and J. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. 102 (2005), no. 17, 5949–5952,  https://doi.org/10.1073/pnas.0501977102.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D. Dürr, M. Pulvirenti, On the vortex flow in bounded domains, Comm. Math. Phys. 85, 265–273 (1982),  https://doi.org/10.1007/BF01254459.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    F. Flandoli, Weak vorticity formulation of 2D Euler equations with white noise initial condition, arXiv:1707.08068 (2017), to appear on Comm. PDEs.
  16. 16.
    F. Gancedo, Existence for the \(\alpha \)-patch model and the QG sharp front in Sobolev spaces, Adv. Math. 217 (2008), no. 6, 2569–2598,  https://doi.org/10.1016/j.aim.2007.10.010.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    C. Geldhauser, M. Romito, Limit theorems and fluctuations for point vortices of generalized Euler equations, arXiv:1810.12706 (2018).
  18. 18.
    C. Geldhauser, M. Romito, Point vortices for inviscid generalized surface quasi-geostrophic models, arXiv:1812.05166 (2018).
  19. 19.
    Z. Hassainia, T. Hmidi, On the V-states for the generalized quasi-geostrophicequations, Comm. Math. Phys. 337 (2015), no. 1, 321–377,  https://doi.org/10.1007/s00220-015-2300-5.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    I.M. Held, R.T. Pierrehumbert, S.T. Garner, K.L. Swanson, Surface quasi-geostrophic dynamics, J. Fluid Mech. 282 (1995), 1–20,  https://doi.org/10.1017/S0022112095000012.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    P. Isett, V. Vicol, Hölder Continuous Solutions of Active Scalar Equations, Ann. PDE 1 (2015), no. 1, Art. 2, 77 pp,  https://doi.org/10.1007/s40818-015-0002-0.
  22. 22.
    A. Kiselev, L. Ryzhik, Y. Yao, A. Zlatoš, Finite time singularity for themodified SQG patch equation, Ann. of Math. 184 (2016) no. 3, 909–948,  https://doi.org/10.4007/annals.2016.184.3.7.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    A. Kiselev, Y. Yao, A. Zlatoš, Local regularity for the modified SQG patch equation, arXiv:1508.07611 (2015).
  24. 24.
    A. J. Majda, A. L. Bertozzi, Vorticity and incompressible flow, Cambridge Univ. Press, 2002,  https://doi.org/10.1017/S0022112003216578.zbMATHGoogle Scholar
  25. 25.
    F. Marchand, Existence and regularity of weak solutions to the quasi-geostrophic equations in the spaces \(L^p\) or \(\dot{H}^{-1/2}\), Commun. Math. Phys. 277 (2008), 45–67,  https://doi.org/10.1007/s00220-007-0356-6.CrossRefzbMATHGoogle Scholar
  26. 26.
    C. Marchioro, M. Pulvirenti, Mathematical theory of incompressible nonviscous fluids, volume 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994,  https://doi.org/10.1007/978-1-4612-4284-0.
  27. 27.
    A. Nahmod, N. Pavlovic, G. Staffilani, N. Totz, Global flows with invariant measures for the inviscid modified SQG equation Stoch PDE: Anal Comp. 6 (2018), no. 2, 184–210,  https://doi.org/10.1007/s40072-017-0106-5.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    H. Q. Nguyen, Global weak solutions for generalized SQG in bounded domains, Anal. PDE 11 (2018), no. 4, 1029–1047,  https://doi.org/10.2140/apde.2018.11.1029. results for msqg in general domainsMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    J. Pedlosky. Geophysical Fluid Dynamics. Second Edition, Springer-Verlag, New York, 1987,  https://doi.org/10.1007/978-1-4612-4650-3.
  30. 30.
    S. Resnick, Dynamical Problems in Nonlinear Advective Partial Differential Equations, Ph. D. thesis University of Chicago, available at https://search.proquest.com/docview/304242616?accountid=14527.
  31. 31.
    J. L. Rodrigo, On the evolution of sharp fronts for the quasi-geostrophic equation, Comm. Pure Appl. Math. 58 (2005), no. 6, 821–866,  https://doi.org/10.1002/cpa.20059.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    S. Schochet, The weak vorticity formulation of the 2-d euler equations and concentration-cancellation, Communications in Partial Differential Equations 20 (1995), no. 5–6, 1077–1104,  https://doi.org/10.1080/03605309508821124.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    K. S. Smith, G. Boccaletti, C. C. Henning, I. Marinov, C. Y. Tam, I. M. Held, G. K. Vallis, Turbulent diffusion in the geostrophic inverse cascade, J. Fluid Mech. 469 (202), 13–48,  https://doi.org/10.1017/S002211200200176.
  34. 34.
    J. Wu, Solutions of the 2D quasi-geostrophic equation in Hölder spaces, Nonlinear Analysis 62 (2005), 579–594,  https://doi.org/10.1016/j.na.2005.03.053.MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Scuola Normale Superiore di Pisa Classe di ScienzePisaItaly
  2. 2.Department of MathematicsTU DarmstadtDarmstadtGermany

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