Global Solutions for the Generalized SQG Patch Equation

  • Diego Córdoba
  • Javier Gómez-Serrano
  • Alexandru D. IonescuEmail author


We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter \({\alpha \in (1,2)}\). The cases \({\alpha = 0}\) and \({\alpha = 1}\) correspond to 2d Euler and SQG respectively, and our choice of the parameter \({\alpha}\) results in a velocity more singular than in the SQG case. Our main result concerns the global stability of the half-plane patch stationary solution, under small and localized perturbations. Our theorem appears to be the first construction of stable global solutions for the gSQG-patch equations. The only other nontrivial global solutions known so far in the patch setting are the so-called V-states, which are uniformly rotating and periodic in time solutions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alazard, T., Burq, N., Zuily, C.: On the water-wave equations with surface tension. Duke Math. J. 158(3), 413–499 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alazard, T., Burq, N., Zuily, C.: On the Cauchy problem for gravity water waves. Invent. Math. 198, 71–163 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alazard, T., Delort, J.M.: Sobolev estimates for two dimensional gravity water waves. Astérisque 374, viii+241 pp, 2015Google Scholar
  4. 4.
    Buckmaster, T., Shkoller, S., Vicol, V.: Nonuniqueness of weak solutions to the SQG equation. Arxiv preprint arXiv:1610.00676, 2016
  5. 5.
    Castro, A., Córdoba, D., Gómez-Serrano, J.: Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations. Duke Math. J. 165(5), 935–984 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Castro, A., Córdoba, D., Gómez-Serrano, J.: Global smooth solutions for the inviscid SQG equation. Arxiv preprint. arXiv:1603.03325, 2016
  7. 7.
    Castro, A., Córdoba, D., Gómez-Serrano, J.: Uniformly rotating analytic global patch solutions for active scalars. Ann. PDE 2(1), 1–34 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Castro, A., Córdoba, D., Gómez-Serrano, J.: Uniformly rotating smooth solutions for the incompressible 2D Euler equations. Arxiv preprint. arXiv:1612.08964, 2016
  9. 9.
    Chae, D., Constantin, P., Córdoba, D., Gancedo, F., Wu, J.: Generalized surface quasi-geostrophic equations with singular velocities. Commun. Pure Appl. Math. 65(8), 1037–1066 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series, 41. Princeton University Press, Princeton, 1993Google Scholar
  11. 11.
    Constantin, P., Lai, M.-C., Sharma, R., Tseng, Y.-H., Wu, J.: New numerical results for the surface quasi-geostrophic equation. J. Sci. Comput. 50(1), 1–28 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Constantin, P., Majda, A.J., Tabak, E.: Formation of strong fronts in the \(2\)-D quasigeostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cordoba, D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. Math. (2) 148(3), 1135–1152, 1998Google Scholar
  14. 14.
    Cordoba, D., Fefferman, C.: Growth of solutions for QG and 2D Euler equations. J. Am. Math. Soc. 15(3), 665–670 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Córdoba, D., Fontelos, M.A., Mancho, A.M., Rodrigo, J.L.: Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA 102(17), 5949–5952 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Deem, G.S., Zabusky, N.J.: Vortex waves: Stationary "V-states", interactions, recurrence, and breaking. Phys. Rev. Lett. 40(13), 859–862 (1978)ADSCrossRefGoogle Scholar
  17. 17.
    Delort, J.M.: Global existence and asymptotic behavior for the quasilinear Klein-Gordon equation with small data in dimension 1. Ann. Sci. École Norm. Sup. 34, 1–61 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Deng, J., Hou, T.Y., Li, R., Yu, X.: Level set dynamics and the non-blowup of the 2D quasi-geostrophic equation. Methods Appl. Anal. 13(2), 157–180 (2006)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Deng, Y., Ionescu, A.D., Pausader, B.: The Euler–Maxwell system for electrons: global solutions in 2d. Arch. Ration. Mech. Anal. 225, 771–871 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Deng Y., Ionescu A.D., Pausader B., Pusateri F.: Global solutions of the gravity-capillary water-wave system in 3 dimensions. Acta Math. 219, 213–402 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dritschel, D.G.: An exact steadily rotating surface quasi-geostrophic elliptical vortex. Geophys. Astrophys. Fluid Dyn. 105(4–5), 368–376 (2011)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Elcrat, A., Fornberg, B., Miller, K.: Stability of vortices in equilibrium with a cylinder. J. Fluid Mech. 544, 53–68 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gancedo, F.: Existence for the \(\alpha \)-patch model and the QG sharp front in Sobolev spaces. Adv. Math. 217(6), 2569–2598 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gancedo, F., Strain, R.M.: Absence of splash singularities for surface quasi-geostrophic sharp fronts and the muskat problem. Proc. Natl. Acad. Sci. 111(2), 635–639 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Germain, P., Masmoudi, N., Shatah, J.: Global solutions for 3d quadratic Schrödinger equations. Int. Math. Res. Not. 414–432, 2009 (2009)zbMATHGoogle Scholar
  26. 26.
    Germain, P., Masmoudi, N., Shatah, J.: Global solutions for the gravity water waves equation in dimension 3. Ann. Math. 2(175), 691–754 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Guo, Y., Ionescu, A.D., Pausader, B.: Global solutions of the Euler-Maxwell two-fluid system in 3d. Ann. Math. 2(183), 377–498 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gustafson, S., Nakanishi, K., Tsai, T.: Scattering for the Gross-Pitaevsky equation in 3 dimensions. Commun. Contemp. Math. 11, 657–707 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hassainia, Z., Hmidi, T.: On the V-states for the generalized quasi-geostrophic equations. Comm. Math. Phys. 337(1), 321–377 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Held, I.M., Pierrehumbert, R.T., Garner, S.T., Swanson, K.L.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ionescu, A.D., Pausader, B.: The Euler-Poisson system in 2d: global stability of the constant equilibrium solution. Int. Math. Res. Not. 2013, 761–826 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ionescu, A.D., Pusateri, F.: Global regularity for 2D water waves with surface tension. Memoirs of the American Mathematical Society, Vol. 256, Number 1227. American Mathematical Society, Providence, RI, 2018Google Scholar
  33. 33.
    Ionescu, A.D., Pusateri, F.: Global solutions for the gravity water waves system in 2D. Invent. Math. 199, 653–804 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ionescu, A.D., Pusateri, F.: Global analysis of a model for capillary water waves in two dimensions. Commun. Pure Appl. Math. 69, 2015–2071 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kiselev, A., Nazarov, F.: A simple energy pump for the surface quasi-geostrophic equation. Nonlinear Partial Differential Equations, Abel Symposia, Vol. 7 (Eds. Holden H. and Karlsen K. H.) Springer, Berlin, 175–179, 2012Google Scholar
  36. 36.
    Kiselev, A., Ryzhik, L., Yao, Y., Zlatoš, A.: Finite time singularity formation for the modified SQG patch equation. Ann. Math. 2(184), 909–948 (2016)CrossRefzbMATHGoogle Scholar
  37. 37.
    Klainerman, S.: Uniform decay estimates and the Lorentz invariance of the classical wave equation. Commun. Pure Appl. Math. 38(3), 321–332 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Klainerman, S.: The null condition and global existence to nonlinear wave equations. In Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), Volume 23 of Lectures in Appl. Math. Amer. Math. Soc., Providence, RI, pp. 293–326, 1986Google Scholar
  39. 39.
    Luzzatto-Fegiz, P., Williamson, C.H.K.: An efficient and general numerical method to compute steady uniform vortices. J. Comput. Phys. 230(17), 6495–6511 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Marchand, F.: 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(1), 45–67 (2008)ADSCrossRefzbMATHGoogle Scholar
  41. 41.
    Nahmod, A., Pavlovic, N., Staffilani, G., Totz, N.: Global flows with invariant measures for the inviscid modified SQG equations. ArXiv preprint arXiv:1705.01890, 2017
  42. 42.
    Resnick, S.G.: Dynamical problems in non-linear advective partial differential equations. PhD thesis, University of Chicago, Department of Mathematics, 1995Google Scholar
  43. 43.
    Rodrigo, J.L.: On the evolution of sharp fronts for the quasi-geostrophic equation. Commun. Pure Appl. Math. 58(6), 821–866 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Saffman, P., Szeto, R.: Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23(12), 2339–2342 (1980)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Scott, R.K.: A scenario for finite-time singularity in the quasigeostrophic model. J. Fluid Mech. 687, 492–502 (2011)ADSCrossRefzbMATHGoogle Scholar
  46. 46.
    Scott, R.K., Dritschel, D.G.: Numerical simulation of a self-similar cascade of filament instabilities in the surface quasigeostrophic system. Phys. Rev. Lett. 112, 144505 (2014)ADSCrossRefGoogle Scholar
  47. 47.
    Shatah, J.: Normal forms and quadratic nonlinear Klein-Gordon equations. Commun. Pure Appl. Math. 38(5), 685–696 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Simon, J.: A wave operator for a nonlinear Klein-Gordon equation. Lett. Math. Phys. 7, 387–398 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Wu, H.M., Overman II, E.A., Zabusky, N.J.: Steady-state solutions of the Euler equations in two dimensions: rotating and translating \(V\)-states with limiting cases. I. Numerical algorithms and results. J. Comput. Phys. 53(1), 42–71, 1984Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Ciencias MatemáticasMadridSpain
  2. 2.Princeton UniversityPrincetonUSA

Personalised recommendations