Eventual regularization for the slightly supercritical quasi-geostrophic equation
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 2, p. 693-704
We prove that weak solutions of the slightly supercritical quasi-geostrophic equation become smooth for large time. The proof uses ideas from a recent article of Caffarelli and Vasseur and is based on an argument in the style of De Giorgi.
Dans cet article, nous montrons que les solutions faibles de l'équation quasi-géostrophique légèrement sur-critique deviennent régulières en temps grand. La démonstration utilise des idées d'un article récent de Caffarelli et Vasseur et repose sur un argument de type de De Giorgi.
@article{AIHPC_2010__27_2_693_0,
     author = {Silvestre, Luis},
     title = {Eventual regularization for the slightly supercritical quasi-geostrophic equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {2},
     year = {2010},
     pages = {693-704},
     doi = {10.1016/j.anihpc.2009.11.006},
     zbl = {1187.35186},
     mrnumber = {2595196},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_2_693_0}
}
Silvestre, Luis. Eventual regularization for the slightly supercritical quasi-geostrophic equation. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 2, pp. 693-704. doi : 10.1016/j.anihpc.2009.11.006. http://www.numdam.org/item/AIHPC_2010__27_2_693_0/

[1] Luis Caffarelli, Luis Silvestre, An extension problem related to the fractional laplacian, Comm. Partial Differential Equations 32 no. 7–9 (2007), 1245-1260 | MR 2354493 | Zbl 1143.26002

[2] Luis Caffarelli, Alexis Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., in press | MR 2680400

[3] P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincare Anal. Non Lineaire 25 no. 6 (2008), 1103-1110 | Numdam | MR 2466323 | Zbl 1149.76052

[4] P. Constantin, J. Wu, Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations, Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009), 159-180 | Numdam | MR 2483817 | Zbl 1163.76010

[5] Peter Constantin, Jiahong Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal. 30 no. 5 (1999), 937-948 | MR 1709781 | Zbl 0957.76093

[6] Antonio Córdoba, Diego Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 no. 3 (2004), 511-528 | MR 2084005 | Zbl 1309.76026

[7] A. Kiselev, F. Nazarov, A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 no. 3 (2007), 445-453 | MR 2276260 | Zbl 1121.35115