Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation

Journal of Mathematical Analysis and Applications

Volumn 339, Issue 1, 2008, Pages 359-371

  Yu, Xinwei ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Quasi geostrophic equation; Regularity conditions; Super critically dissipative

Indexed keywords

EID: 35248823548     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2007.06.064     Document Type: Article

References (30)

1. Caffarelli L., and Silvestre L. An extension problem related to the fractional Laplacian. preprint, arXiv: (August 2006). math.AP/0608640 v1
2. Caffarelli L., and Vasseur A. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. preprint, arXiv: (August 2006). math.AP/0608447 v1
3. Chae D. The quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinearity 16 (2003) 479-495
4. Chae D. On the regularity conditions for the dissipative quasi-geostrophic equations. SIAM J. Math. Anal. 37 (2006) 1649-1656
5. Chae D., and Lee J. Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Comm. Math. Phys. 233 (2003) 297-311
6. Constantin P., Cordoba D., and Wu J. On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50 (2001) 97-107
7. Constantin P., Majda A.J., and Tabak E. Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7 (1994) 1495-1533
8. Constantin P., Nie Q., and Schörghofer N. Nonsingular surface quasi-geostrophic flow. Phys. Lett. A 241 3 (1998) 168-172
9. Constantin P., and Wu J. Behavior of solutions of 2d quasi-geostrophic equations. SIAM J. Math. Anal. 30 (1999) 937-948
10. Cordoba A., and Cordoba D. A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249 (2004) 511-528
11. Cordoba D. Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. 148 3 (1998) 1135-1152
12. Cordoba D., and Fefferman C. Growth of solutions for QG and 2D Euler equations. J. Amer. Math. Soc. 15 3 (2002) 665-670 (electronic)
13. Jian Deng, Thomas Y. Hou, Ruo Li, Xinwei Yu, Level set dynamics and the non-blowup of the 2D quasi-geostrophic equation, Methods Appl. Anal. (2006), in press
14. Dong H. Higher regularity for the critical and super-critical dissipative quasi-geostrophic equations. preprint, arXiv: (2007). math/0701826
15. Dong H., and Du D. Global well-posedness and a decay estimate for the critical quasi-geostrophic equation. preprint, arXiv: (2007). math/0701828
16. Efimov A.V. Linear methods for the approximation of continuous periodic functions. Mat. Sb. 54 (1961) 51-90 (in Russian)
17. Gala S., and Lahmar-Benbernou A. Dissipative quasi-geostrophic equations with initial data. preprint, arXiv: (July 2005). math.AP/0507492 v1
18. Ju N. Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Comm. Math. Phys. 251 2 (2004) 365-376
19. Ju N. The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Comm. Math. Phys. 255 1 (2005) 161-181
20. Ju N. On the two-dimensional quasi-geostrophic equations. Indiana Univ. Math. J. 54 3 (2005) 897-926
21. Ju N. Geometric constrains for global regularity of 2D quasi-geostrophic flows. J. Differential Equations 226 (2006) 54-79
22. Ju N. Global solutions to the two-dimensional quasi-geostrophic equation with critical or super-critical dissipation. Math. Ann. 334 (2006) 627-642
23. Kiselev A., Nazarov F., and Volberg A. Global well-posedness for the critical 2d dissipative quasi-geostrophic equation. Invent. Math. 167 (2007) 445-453
24. Miura H. Dissipative quasi-geostrophic equation for large initial data in the critical Sobolev space. Comm. Math. Phys. 267 (2006) 141-157
25. Stefanov A. Global well-posedness for the 2d quasi-geostrophic equation in a critical Besov space. preprint, arXiv: (July 2006). math.AP/0607320 v1
26. p data. Electron. J. Differential Equations 2001 56 (2001) 1-13
27. Wu J. Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal. 36 (2005) 1014-1030
28. Wu J. Solutions of the 2D quasi-geostrophic equation in Hölder spaces. Nonlinear Anal. 62 (2005) 579-594
29. Wu J. The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation. Nonlinearity 18 (2005) 139-154
30. Zygmund A. Trigonometric Series, vol. 1 (1959), Cambridge Univ. Press