Note on the well-posedness of a slightly supercritical surface quasi-geostrophic equation

Journal of Differential Equations

Volumn 253, Issue 2, 2012, Pages 795-813

  Xue, Liutang ^a   Zheng, Xiaoxin ^a  

^a RESEARCH CENTER OF LASER FUSION   (China)

Author keywords

Modulus of continuity; Nonlocal maximum principle; Slightly supercritical equation; Surface quasi geostrophic equation

Indexed keywords

EID: 84860610382     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2012.04.003     Document Type: Article

References (19)

1. Bahouri H., Chemin J.-Y., Danchin R. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren Math. Wiss. 2011, vol. 343. Springer-Verlag.
2. Caffarelli L., Vasseur V. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equations. Ann. of Math. 2010, 171(3):1903-1930.
3. Castro A., Córdoba D., Gancedo F., Orive R. Incompressible flow in porous media with fractional diffusion. Nonlinearity 2009, 22(8):1791-1815.
4. Chae D., Constantin P., Wu J. Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations. Arch. Ration. Mech. Anal. 2011, 202(1):35-62.
5. Constantin P., Iyer G., Wu J. Global regularity for a modified critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 2008, 57:2681-2692.
6. Constantin P., Majda A.J., Tabak E. Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 1994, 7:1495-1533.
7. Constantin P., Wu J. Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 1999, 30:937-948.
8. Constantin P., Wu J. Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 2008, 25(6):1103-1110.
9. P. Constantin, V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., in press. arxiv:1110.0179.
10. Dabkowski M. Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation. Geom. Funct. Anal. 2011, 21(1):1-13.
11. M. Dabkowski, A. Kiselev, V. Vicol, Global well-posedness for a slightly supercritical surface quasi-geostrophic equation, Nonlinearity, in press. arxiv:1106.2137.
12. Friedlander S., Rusin W., Vicol V. On the supercritically diffusive magneto-geostrophic equations arxiv:1110.1029.
13. Kiselev A. Nonlocal maximum principle for active scalars. Adv. Math. 2011, 227(5):1806-1826.
14. Kiselev A., Nazarov F., Volberg A. Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 2007, 167:445-453.
15. Kiselev A., Nazarov F. A variation on a theme of Caffarelli and Vasseur. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 2010, 370:58-72.
16. Miao C., Xue L. On the regularity of a class of generalized quasi-geostrophic equations. J. Differential Equations 2011, 251:2789-2821.
17. Miao C., Xue L. Global well-posedness for a modified critical dissipative quasi-geostrophic equation. J. Differential Equations 2012, 252:792-818.
18. S. Resnick, Dynamical problems in nonlinear advective partial differential equations, Ph.D. thesis, University of Chicago, 1995.
19. Yu X. Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation. J. Math. Anal. Appl. 2008, 339:359-371.