New numerical results for the surface quasi-geostrophic equation

Journal of Scientific Computing

Volumn 50, Issue 1, 2012, Pages 1-28

  Constantin, Peter ^a   Lai, Ming Chih ^b   Sharma, Ramjee ^c   Tseng, Yu Hou ^b   Wu, Jiahong ^d  

^a UNIVERSITY OF CHICAGO   (United States)

^b NATIONAL CHIAO TUNG UNIVERSITY   (Taiwan)

^c GEORGIA PERIMETER COLLEGE   (United States)

^d OKLAHOMA STATE UNIVERSITY   (United States)

Author keywords

Global regularity; Parallel computation; Surface quasi geostrophic equation

Indexed keywords

ASYMPTOTICS; CLASSICAL SOLUTIONS; FINITE TIME SINGULARITY; GLOBAL REGULARITY; KEY PARAMETERS; NON-LINEARITY; NUMERICAL COMPUTATIONS; NUMERICAL RESULTS; PARALLEL COMPUTATION; PARALLELIZING; SURFACE QUASI-GEOSTROPHIC EQUATION; SURFACE QUASI-GEOSTROPHIC EQUATIONS; TIME INTERVAL;

ASTROPHYSICS;

CONTROL NONLINEARITIES;

EID: 84855663746     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-011-9471-9     Document Type: Article

References (73)

1. H. Abidi T. Hmidi 2008 On the global well-posedness of the critical quasi-geostrophic equation SIAM J. Math. Anal. 40 167 185 10.1137/070682319 1157.76054 2403316
2. W. Blumen 1978 Uniform potential vorticity flow, Part I. Theory of wave interactions and two-dimensional turbulence J. Atmos. Sci. 35 774 783 10.1175/1520-0469(1978)035<0774:UPVFPI>2.0.CO;2
3. L. Caffarelli L. Silvestre 2007 An extension problem related to the fractional Laplacian Commun. Partial Differ. Equ. 32 1245 1260 10.1080/03605300600987306 1143.26002 2354493
4. L. Caffarelli A. Vasseur 2010 Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation Ann. Math. 171 1903 1930 10.4007/annals.2010.171.1903 1204.35063 2680400
5. J. Carrillo L. Ferreira 2008 The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations Nonlinearity 21 1001 1018 10.1088/0951-7715/21/5/006 1136.76052 2412324
6. D. Chae 2003 The quasi-geostrophic equation in the Triebel-Lizorkin spaces Nonlinearity 16 479 495 10.1088/0951-7715/16/2/307 1029.35006 1958612
7. D. Chae 2006 On the regularity conditions for the dissipative quasi-geostrophic equations SIAM J. Math. Anal. 37 1649 1656 10.1137/040616954 1141.76010 2215601 (Pubitemid 44422778)
8. D. Chae 2008 The geometric approaches to the possible singularities in the inviscid fluid flows J. Phys. A 41 365501 2447877 11 p.
9. D. Chae J. Lee 2003 Global well-posedness in the super-critical dissipative quasi-geostrophic equations Commun. Math. Phys. 233 297 311 1019.86002 1962043 (Pubitemid 36256673)
10. Q. Chen C. Miao Z. Zhang 2007 A new Bernstein's inequality and the 2D dissipative quasi-geostrophic equation Commun. Math. Phys. 271 821 838 10.1007/s00220-007-0193-7 1142.35069 2291797
11. Q. Chen Z. Zhang 2007 Global well-posedness of the 2D critical dissipative quasi-geostrophic equation in the Triebel-Lizorkin spaces Nonlinear Anal. 67 1715 1725 10.1016/j.na.2006.08.011 1115.76081 2326024 (Pubitemid 46634836)
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) (Pubitemid 44418415)
13. P. Constantin D. Córdoba J. Wu 2001 On the critical dissipative quasi-geostrophic equation Indiana Univ. Math. J. 50 97 107 0989.86004 1855665
14. P. Constantin G. Iyer J. Wu 2008 Global regularity for a modified critical dissipative quasi-geostrophic equation Indiana Univ. Math. J. 57 2681 2692 10.1512/iumj.2008.57.3629 1159.35059 2482996
15. P. Constantin A. Majda E. Tabak 1994 Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar Nonlinearity 7 1495 1533 10.1088/0951-7715/7/6/001 0809.35057 1304437
16. P. Constantin Q. Nie N. Schörghofer 1998 Nonsingular surface quasi-geostrophic flow Phys. Lett. A 241 168 172 10.1016/S0375-9601(98)00108-X 0974.76512 1613907 (Pubitemid 128391754)
17. P. Constantin J. Wu 1999 Behavior of solutions of 2D quasi-geostrophic equations SIAM J. Math. Anal. 30 937 948 10.1137/S0036141098337333 0957.76093 1709781 (Pubitemid 129486164)
18. P. Constantin J. Wu 2008 Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25 1103 1110 10.1016/j.anihpc.2007. 10.001 1149.76052 2466323
19. P. Constantin J. Wu 2009 Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equation Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 159 180 10.1016/j.anihpc.2007.10. 002 1163.76010 2483817
20. D. Córdoba 1998 Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation Ann. Math. 148 1135 1152 10.2307/121037 0920.35109 (Pubitemid 128386063)
21. A. Córdoba D. Córdoba 2004 A maximum principle applied to quasi-geostrophic equations Commun. Math. Phys. 249 511 528 10.1007/s00220-004- 1055-1 02158321 (Pubitemid 39216537)
22. D. Córdoba Ch. Fefferman 2001 Behavior of several two-dimensional fluid equations in singular scenarios Proc. Natl. Acad. Sci. USA 98 4311 4312 10.1073/pnas.061038998 0965.35133 1819132 (Pubitemid 32294972)
23. D. Córdoba Ch. Fefferman 2002 Scalars convected by a two-dimensional incompressible flow Commun. Pure Appl. Math. 55 255 260 10.1002/cpa.3007 1019.76013 (Pubitemid 37244221)
24. D. Córdoba Ch. Fefferman 2002 Growth of solutions for QG and 2D Euler equations J. Am. Math. Soc. 15 665 670 10.1090/S0894-0347-02-00394-6 1013.76011
25. D. Córdoba M. Fontelos A. Mancho J. Rodrigo 2005 Evidence of singularities for a family of contour dynamics equations Proc. Natl. Acad. Sci. USA 102 5949 5952 10.1073/pnas.0501977102 1135.76315 2141918 (Pubitemid 40594235)
26. J. Deng T.Y. Hou R. Li X. Yu 2006 Level set dynamics and the non-blowup of the 2D quasi-geostrophic equation Methods Appl. Anal. 13 157 180 1173.76006 2381544
27. B. Dong Z. Chen 2006 Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation Nonlinearity 19 2919 2928 10.1088/0951-7715/19/12/011 1109.76063 2273766
28. H. Dong D. Du 2008 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 10.3934/dcds.2008.21.1095 1141.35436 2399451
29. H. Dong D. Li 2008 Finite time singularities for a class of generalized surface quasi-geostrophic equations Proc. Am. Math. Soc. 136 2555 2563 10.1090/S0002-9939-08-09328-3 1143.35084 2390526
30. H. Dong D. Li 2008 Spatial analyticity of the solutions to the subcritical dissipative quasi-geostrophic equations Arch. Ration. Mech. Anal. 189 131 158 10.1007/s00205-007-0100-6 1157.35456 2403602
31. H. Dong N. Pavlovic 2009 A regularity criterion for the dissipation quasi-geostrophic equation Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 1607 1619 10.1016/j.anihpc.2008.08.001 1176.35133 2566702
32. H. Dong N. Pavlovic 2009 Regularity criteria for the dissipative quasi-geostrophic equations in Holder spaces Commun. Math. Phys. 290 801 812 10.1007/s00220-009-0756-x 1185.35187 2525640
33. Gill, A.E.: Atmosphere-Ocean Dynamics. Academic Press, San Diego (1982)
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)
35. I. Held R. Pierrehumbert S. Garner K. Swanson 1995 Surface quasi-geostrophic dynamics J. Fluid Mech. 282 1 20 10.1017/S0022112095000012 0832.76012 1312238 (Pubitemid 26463360)
36. T. Hmidi S. Keraani 2007 Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces Adv. Math. 214 618 638 10.1016/j.aim.2007.02.013 1119.76070 2349714
37. T.Y. Hou R. Li 2007 Computing nearly singular solutions using pseudo-spectral methods J. Comput. Phys. 226 379 397 10.1016/j.jcp.2007.04.014 05202143 2356364
38. N. Ju 2005 The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations Commun. Math. Phys. 255 161 181 10.1007/s00220-004-1256-7 1088.37049 2123380 (Pubitemid 40900257)
39. N. Ju 2006 Geometric constrains for global regularity of 2D quasi-geostrophic flows J. Differ. Equ. 226 54 79 10.1016/j.jde.2006.03.010 1098.35046 2232429 (Pubitemid 43714142)
40. Kiselev, A.: Some recent results on the critical surface quasi-geostrophic equation: a review, preprint
41. A. Kiselev F. Nazarov A. Volberg 2007 Global well-posedness for the critical 2D dissipative quasi-geostrophic equation Invent. Math. 167 445 453 10.1007/s00222-006-0020-3 1121.35115 2276260
42. Lemarie-Rieusset, P.-G.: Recent Developments in the Navier-Stokes Problem. Chapman & Hall/CRC, Boca Raton (2002)
43. D. Li 2009 Existence theorems for the 2D quasi-geostrophic equation with plane wave initial conditions Nonlinearity 22 1639 1651 10.1088/0951-7715/22/7/ 008 1177.35170 2519682
44. D. Li J. Rodrigo 2009 Blow up for the generalized surface quasi-geostrophic equation with supercritical dissipation Commun. Math. Phys. 286 111 124 10.1007/s00220-008-0585-3 1172.86301 2470925
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)
46. A. Majda E. Tabak 1996 A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow Physica D 98 515 522 10.1016/0167-2789(96)00114-5 0899.76105 1422288 (Pubitemid 126687472)
47. F. Marchand 2006 Propagation of Sobolev regularity for the critical dissipative quasi-geostrophic equation Asymptot. Anal. 49 275 293 1115.35031 2270894 (Pubitemid 44434394)
48. -1/2 Commun. Math. Phys. 277 45 67 10.1007/s00220-007-0356-6 1134.35025 2357424
49. F. Marchand 2008 Weak-strong uniqueness criteria for the critical quasi-geostrophic equation Physica D 237 1346 1351 10.1016/j.physd.2008.03.011 1143.76339 2454593
50. F. Marchand P.G. Lemarié-Rieusset 2005 Solutions auto-similaires non radiales pour l'équation quasi-géostrophique dissipative critique C. R. Math. Acad. Sci. Paris 341 535 538 1155.76320 2181389 (Pubitemid 41600439)
51. H. Miura 2006 Dissipative quasi-geostrophic equation for large initial data in the critical Sobolev space Commun. Math. Phys. 267 141 157 10.1007/s00220-006-0023-3 1113.76029 2238907 (Pubitemid 44230342)
52. K. Ohkitani M. Yamada 1997 Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow Phys. Fluids 9 876 882 10.1063/1.869184 1185.76841 1437554
53. Okitani, K., Sakajo, T.: Oscillatory damping in long-time evolution of the surface quasi-geostrophic equations with generalized viscosity: a numerical study, preprint
54. Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1987)
55. Resnick, S.: Dynamical problems in nonlinear advective partial differential equations. Ph.D. thesis, University of Chicago (1995)
56. J. Rodrigo 2004 The vortex patch problem for the surface quasi-geostrophic equation Proc. Natl. Acad. Sci. USA 101 2684 2686 10.1073/pnas.0308158101 1063.76009 2036969 (Pubitemid 38327670)
57. J. Rodrigo 2005 On the evolution of sharp fronts for the quasi-geostrophic equation Commun. Pure Appl. Math. 58 821 866 10.1002/cpa.20059 1073.35006 2142632 (Pubitemid 40763809)
58. M. Schonbek T. Schonbek 2003 Asymptotic behavior to dissipative quasi-geostrophic flows SIAM J. Math. Anal. 35 357 375 10.1137/S0036141002409362 1126.76386 2001105
59. M. Schonbek T. Schonbek 2005 Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows Discrete Contin. Dyn. Syst. 13 1277 1304 10.3934/dcds.2005.13.1277 1091.35070 2166670 (Pubitemid 41730770)
60. J. Wu 1997 Quasi-geostrophic-type equations with initial data in Morrey spaces Nonlinearity 10 1409 1420 10.1088/0951-7715/10/6/002 0906.35078 1483549 (Pubitemid 127362600)
61. J. Wu 1997 Inviscid limits and regularity estimates for the solutions of the 2-D dissipative quasi-geostrophic equations Indiana Univ. Math. J. 46 1113 1124 10.1512/iumj.1997.46.1275 0909.35111 1631560 (Pubitemid 127374560)
62. p data Electron. J. Differ. Equ. 2001 1 13
63. J. Wu 2002 The quasi-geostrophic equation and its two regularizations Commun. Partial Differ. Equ. 27 1161 1181 10.1081/PDE-120004898 1012.35067
64. J. Wu 2004 The generalized incompressible Navier-Stokes equations in Besov spaces Dyn. Partial Differ. Equ. 1 381 400 1075.35043 2127578
65. J. Wu 2004/2005 Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces SIAM J. Math. Anal. 36 1014 1030 10.1137/ S0036141003435576
66. J. Wu 2005 The quasi-geostrophic equation with critical or supercritical dissipation Nonlinearity 18 139 154 10.1088/0951-7715/18/1/008 1067.35002 2109471 (Pubitemid 40097523)
67. J. Wu 2005 Solutions of the 2-D quasi-geostrophic equation in Hölder spaces Nonlinear Anal. 62 579 594 10.1016/j.na.2005.03.053 1116.35348 2149903 (Pubitemid 40822109)
68. J. Wu 2006 Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces Commun. Math. Phys. 263 803 831 10.1007/s00220-005-1483-6 1104.35037 (Pubitemid 43473641)
69. J. Wu 2007 Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation Nonlinear Anal. 67 3013 3036 10.1016/j.na.2006.09.050 1122.76014 2347594 (Pubitemid 47187931)
70. X. Yu 2008 Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation J. Math. Anal. Appl. 339 359 371 10.1016/j.jmaa.2007.06.064 1128.35006 2370657
71. Z. Zhang 2005 Well-posedness for the 2D dissipative quasi-geostrophic equations in the Besov space Sci. China Ser. A 48 1646 1655 10.1360/04ys0210 1217.35148 2203339 (Pubitemid 41804610)
72. Z. Zhang 2007 Global well-posedness for the 2D critical dissipative quasi-geostrophic equation Sci. China Ser. A 50 485 494 10.1007/s11425-007-0002- y 2333748
73. Y. Zhou 2006 Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows Discrete Contin. Dyn. Syst. 14 525 532 10.3934/dcds.2006.14.525 1185.35203 2171725 (Pubitemid 43294305)