Journal of Scientific Computing

, Volume 50, Issue 1, pp 1–28 | Cite as

New Numerical Results for the Surface Quasi-Geostrophic Equation

  • Peter Constantin
  • Ming-Chih Lai
  • Ramjee Sharma
  • Yu-Hou Tseng
  • Jiahong Wu
Article

Abstract

The question whether classical solutions of the surface quasi-geostrophic (SQG) equation can develop finite-time singularities remains open. This paper presents new numerical computations of the solutions to the SQG equation corresponding to several classes of initial data previously proposed by Constantin et al. (Nonlinearity 7:1495–1533, 1994). By parallelizing the serial pseudo-spectral codes through slab decompositions and applying suitable filters, we are able to simulate these solutions with great precision and on large time intervals. These computations reveal detailed finite-time behavior, large-time asymptotics and key parameter dependence of the solutions and provide information for further investigations on the global regularity issue concerning the SQG equation.

Keywords

Global regularity Parallel computation Surface quasi-geostrophic equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abidi, H., Hmidi, T.: On the global well-posedness of the critical quasi-geostrophic equation. SIAM J. Math. Anal. 40, 167–185 (2008) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Blumen, W.: Uniform potential vorticity flow, Part I. Theory of wave interactions and two-dimensional turbulence. J. Atmos. Sci. 35, 774–783 (1978) CrossRefGoogle Scholar
  3. 3.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 171, 1903–1930 (2010) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Carrillo, J., Ferreira, L.: The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations. Nonlinearity 21, 1001–1018 (2008) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chae, D.: The quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinearity 16, 479–495 (2003) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chae, D.: On the regularity conditions for the dissipative quasi-geostrophic equations. SIAM J. Math. Anal. 37, 1649–1656 (2006) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Chae, D.: The geometric approaches to the possible singularities in the inviscid fluid flows. J. Phys. A 41, 365501 (2008), 11 p. MathSciNetGoogle Scholar
  9. 9.
    Chae, D., Lee, J.: Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Commun. Math. Phys. 233, 297–311 (2003) MATHMathSciNetGoogle Scholar
  10. 10.
    Chen, Q., Miao, C., Zhang, Z.: A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation. Commun. Math. Phys. 271, 821–838 (2007) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Chen, Q., Zhang, Z.: Global well-posedness of the 2D critical dissipative quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinear Anal. 67, 1715–1725 (2007) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Constantin, P.: Euler equations, Navier-Stokes equations and turbulence. In: Mathematical Foundation of Turbulent Viscous Flows. Lecture Notes in Math., vol. 1871, pp. 1–43. Springer, Berlin (2006) CrossRefGoogle Scholar
  13. 13.
    Constantin, P., Córdoba, D., Wu, J.: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50, 97–107 (2001) MATHMathSciNetGoogle Scholar
  14. 14.
    Constantin, P., Iyer, G., Wu, J.: Global regularity for a modified critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 57, 2681–2692 (2008) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Constantin, P., Nie, Q., Schörghofer, N.: Nonsingular surface quasi-geostrophic flow. Phys. Lett. A 241, 168–172 (1998) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937–948 (1999) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Constantin, P., Wu, J.: Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, 1103–1110 (2008) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Constantin, P., Wu, J.: Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, 159–180 (2009) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Córdoba, D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. Math. 148, 1135–1152 (1998) CrossRefMATHGoogle Scholar
  21. 21.
    Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249, 511–528 (2004) CrossRefMATHGoogle Scholar
  22. 22.
    Córdoba, D., Fefferman, Ch.: Behavior of several two-dimensional fluid equations in singular scenarios. Proc. Natl. Acad. Sci. USA 98, 4311–4312 (2001) CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Córdoba, D., Fefferman, Ch.: Scalars convected by a two-dimensional incompressible flow. Commun. Pure Appl. Math. 55, 255–260 (2002) CrossRefMATHGoogle Scholar
  24. 24.
    Córdoba, D., Fefferman, Ch.: Growth of solutions for QG and 2D Euler equations. J. Am. Math. Soc. 15, 665–670 (2002) CrossRefMATHGoogle Scholar
  25. 25.
    Córdoba, D., Fontelos, M., Mancho, A., Rodrigo, J.: Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA 102, 5949–5952 (2005) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    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, 157–180 (2006) MATHMathSciNetGoogle Scholar
  27. 27.
    Dong, B., Chen, Z.: Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation. Nonlinearity 19, 2919–2928 (2006) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Dong, H., Du, D.: Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete Contin. Dyn. Syst. 21, 1095–1101 (2008) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Dong, H., Li, D.: Finite time singularities for a class of generalized surface quasi-geostrophic equations. Proc. Am. Math. Soc. 136, 2555–2563 (2008) CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Dong, H., Li, D.: Spatial analyticity of the solutions to the subcritical dissipative quasi-geostrophic equations. Arch. Ration. Mech. Anal. 189, 131–158 (2008) CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Dong, H., Pavlovic, N.: A regularity criterion for the dissipation quasi-geostrophic equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, 1607–1619 (2009) CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Dong, H., Pavlovic, N.: Regularity criteria for the dissipative quasi-geostrophic equations in Holder spaces. Commun. Math. Phys. 290, 801–812 (2009) CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Gill, A.E.: Atmosphere-Ocean Dynamics. Academic Press, San Diego (1982) Google Scholar
  34. 34.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 26. SIAM, Philadelphia (1977) CrossRefMATHGoogle Scholar
  35. 35.
    Held, I., Pierrehumbert, R., Garner, S., Swanson, K.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995) CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Hmidi, T., Keraani, S.: Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces. Adv. Math. 214, 618–638 (2007) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Hou, T.Y., Li, R.: Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226, 379–397 (2007) CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Ju, N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Commun. Math. Phys. 255, 161–181 (2005) CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Ju, N.: Geometric constrains for global regularity of 2D quasi-geostrophic flows. J. Differ. Equ. 226, 54–79 (2006) CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Kiselev, A.: Some recent results on the critical surface quasi-geostrophic equation: a review, preprint Google Scholar
  41. 41.
    Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167, 445–453 (2007) CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Lemarie-Rieusset, P.-G.: Recent Developments in the Navier-Stokes Problem. Chapman & Hall/CRC, Boca Raton (2002) CrossRefMATHGoogle Scholar
  43. 43.
    Li, D.: Existence theorems for the 2D quasi-geostrophic equation with plane wave initial conditions. Nonlinearity 22, 1639–1651 (2009) CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Li, D., Rodrigo, J.: Blow up for the generalized surface quasi-geostrophic equation with supercritical dissipation. Commun. Math. Phys. 286, 111–124 (2009) CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes, vol. 9. Courant Institute of Mathematical Sciences and American Mathematical Society, New York (2003) MATHGoogle Scholar
  46. 46.
    Majda, A., Tabak, E.: A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow. Physica D 98, 515–522 (1996) CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Marchand, F.: Propagation of Sobolev regularity for the critical dissipative quasi-geostrophic equation. Asymptot. Anal. 49, 275–293 (2006) MATHMathSciNetGoogle Scholar
  48. 48.
    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, 45–67 (2008) CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Marchand, F.: Weak-strong uniqueness criteria for the critical quasi-geostrophic equation. Physica D 237, 1346–1351 (2008) CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Marchand, F., Lemarié-Rieusset, P.G.: Solutions auto-similaires non radiales pour l’équation quasi-géostrophique dissipative critique. C. R. Math. Acad. Sci. Paris 341, 535–538 (2005) MATHMathSciNetGoogle Scholar
  51. 51.
    Miura, H.: Dissipative quasi-geostrophic equation for large initial data in the critical Sobolev space. Commun. Math. Phys. 267, 141–157 (2006) CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Ohkitani, K., Yamada, M.: Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow. Phys. Fluids 9, 876–882 (1997) CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    Okitani, K., Sakajo, T.: Oscillatory damping in long-time evolution of the surface quasi-geostrophic equations with generalized viscosity: a numerical study, preprint Google Scholar
  54. 54.
    Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1987) CrossRefMATHGoogle Scholar
  55. 55.
    Resnick, S.: Dynamical problems in nonlinear advective partial differential equations. Ph.D. thesis, University of Chicago (1995) Google Scholar
  56. 56.
    Rodrigo, J.: The vortex patch problem for the surface quasi-geostrophic equation. Proc. Natl. Acad. Sci. USA 101, 2684–2686 (2004) CrossRefMATHMathSciNetGoogle Scholar
  57. 57.
    Rodrigo, J.: On the evolution of sharp fronts for the quasi-geostrophic equation. Commun. Pure Appl. Math. 58, 821–866 (2005) CrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    Schonbek, M., Schonbek, T.: Asymptotic behavior to dissipative quasi-geostrophic flows. SIAM J. Math. Anal. 35, 357–375 (2003) CrossRefMATHMathSciNetGoogle Scholar
  59. 59.
    Schonbek, M., Schonbek, T.: Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Discrete Contin. Dyn. Syst. 13, 1277–1304 (2005) CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    Wu, J.: Quasi-geostrophic-type equations with initial data in Morrey spaces. Nonlinearity 10, 1409–1420 (1997) CrossRefMATHMathSciNetGoogle Scholar
  61. 61.
    Wu, J.: Inviscid limits and regularity estimates for the solutions of the 2-D dissipative quasi-geostrophic equations. Indiana Univ. Math. J. 46, 1113–1124 (1997) CrossRefMATHMathSciNetGoogle Scholar
  62. 62.
    Wu, J.: Dissipative quasi-geostrophic equations with L p data. Electron. J. Differ. Equ. 2001, 1–13 (2001) Google Scholar
  63. 63.
    Wu, J.: The quasi-geostrophic equation and its two regularizations. Commun. Partial Differ. Equ. 27, 1161–1181 (2002) CrossRefMATHGoogle Scholar
  64. 64.
    Wu, J.: The generalized incompressible Navier-Stokes equations in Besov spaces. Dyn. Partial Differ. Equ. 1, 381–400 (2004) MATHMathSciNetGoogle Scholar
  65. 65.
    Wu, J.: Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal. 36, 1014–1030 (2004/2005) CrossRefGoogle Scholar
  66. 66.
    Wu, J.: The quasi-geostrophic equation with critical or supercritical dissipation. Nonlinearity 18, 139–154 (2005) CrossRefMATHMathSciNetGoogle Scholar
  67. 67.
    Wu, J.: Solutions of the 2-D quasi-geostrophic equation in Hölder spaces. Nonlinear Anal. 62, 579–594 (2005) CrossRefMATHMathSciNetGoogle Scholar
  68. 68.
    Wu, J.: Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces. Commun. Math. Phys. 263, 803–831 (2006) CrossRefMATHGoogle Scholar
  69. 69.
    Wu, J.: Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation. Nonlinear Anal. 67, 3013–3036 (2007) CrossRefMATHMathSciNetGoogle Scholar
  70. 70.
    Yu, X.: Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation. J. Math. Anal. Appl. 339, 359–371 (2008) CrossRefMATHMathSciNetGoogle Scholar
  71. 71.
    Zhang, Z.: Well-posedness for the 2D dissipative quasi-geostrophic equations in the Besov space. Sci. China Ser. A 48, 1646–1655 (2005) CrossRefMATHMathSciNetGoogle Scholar
  72. 72.
    Zhang, Z.: Global well-posedness for the 2D critical dissipative quasi-geostrophic equation. Sci. China Ser. A 50, 485–494 (2007) CrossRefMathSciNetGoogle Scholar
  73. 73.
    Zhou, Y.: Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows. Discrete Contin. Dyn. Syst. 14, 525–532 (2006) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Peter Constantin
    • 1
  • Ming-Chih Lai
    • 2
  • Ramjee Sharma
    • 3
  • Yu-Hou Tseng
    • 2
  • Jiahong Wu
    • 4
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of Applied MathematicsNational Chiao Tung UniversityHsinchuTaiwan
  3. 3.Department of MathematicsGeorgia Perimeter CollegeAtlantaUSA
  4. 4.Department of MathematicsOklahoma State UniversityStillwaterUSA

Personalised recommendations