A Regularity Criterion for the Dissipative Quasi-Geostrophic Equations

Dong, Hongjie ; Pavlović, NatašA

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009), p. 1607-1619 / Harvested from Numdam

Détail

Publié le : 2009-01-01

MR 2566702 | Zbl 1176.35133

DOI : https://doi.org/10.1016/j.anihpc.2008.08.001

Bibliographie

[1] L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, preprint.

[2] Chae D., The Quasi-Geostrophic Equation in the Triebel-Lizorkin Spaces, Nonlinearity 16 (2) (2003) 479-495. | MR 1958612 | Zbl 1029.35006

[3] Chae D., On the Regularity Conditions for the Dissipative Quasi-Geostrophic Equations, SIAM J. Math. Anal. 37 (5) (2006) 1649-1656. | MR 2215601 | Zbl 1141.76010

[4] Chae D., Lee J., Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations, Commun. Math. Phys. 233 (2003) 297-311. | MR 1962043 | Zbl 1019.86002

[5] Chemin J.-Y., Théorèmes D'unicité Pour Le Système De Navier-Stokes Tridimensionnel, J. Anal. Math. 77 (1999) 27-50, (in French). | MR 1753481 | Zbl 0938.35125

[6] Chen Q., Miao C., Zhang Z., A New Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation, Commun. Math. Phys. 271 (3) (2007) 821-838. | MR 2291797 | Zbl 1142.35069

[7] Cheskidov A., Shvydkoy R., On the Regularity of Weak Solutions of the 3D Navier-Stokes Equations in ${B}_{\infty ,\infty }^{-1}$, preprint, arXiv: math.AP/0708.3067. | MR 2564471

[8] Constantin P., Cordoba D., Wu J., On the Critical Dissipative Quasi-Geostrophic Equation, Indiana Univ. Math. J. 50 (2001) 97-107. | MR 1855665 | Zbl 0989.86004

[9] Constantin P., Majda A. J., Tabak E., Formation of Strong Fronts in the 2-D Quasigeostrophic Thermal Active Scalar, Nonlinearity 7 (6) (1994) 1495-1533. | MR 1304437 | Zbl 0809.35057

[10] Constantin P., Wu J., Behavior of Solutions of 2D Quasi-Geostrophic Equations, SIAM J. Math. Anal. 30 (1999) 937-948. | MR 1709781 | Zbl 0957.76093

[11] P. Constantin, J. Wu, Hölder continuity of solutions of super-critical dissipative hydrodynamic transport equations, Ann. I. H. Poincaré - AN (2007), doi:10.1016/j.anihpc.2007.10.002. | Numdam | Zbl 1163.76010

[12] P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. I. H. Poincaré - AN (2007), doi:10.1016/j.anihpc.2007.10.001. | Numdam | Zbl 1149.76052

[13] Danchin R., Density-Dependent Incompressible Viscous Fluids in Critical Spaces, Proc. Roy. Soc. Edinburgh Sect. A 133 (6) (2003) 1311-1334. | MR 2027648 | Zbl 1050.76013

[14] Dong B.-Q., Chen Z.-M., A Remark on Regularity Criterion for the Dissipative Quasi-Geostrophic Equations, J. Math. Anal. Appl. (2007) 1212-1217. | MR 2296919 | Zbl 1154.76339

[15] H. Dong, Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness, 2007, submitted for publication.

[16] 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 (4) (2008) 1095-1101. | MR 2399451 | Zbl 1141.35436

[17] H. Dong, D. Li, On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces, 2007, submitted for publication.

[18] Escauriaza L., Seregin G., Šverak V., ${L}^{3,\infty }$-Solutions of the Navier-Stokes Equations and Backward Uniqueness, Russian Math. Surveys 58 (2003). | MR 1992563 | Zbl 1064.35134

[19] Giga Y., Solutions for Semilinear Parabolic Equations in ${L}^{p}$ and Regularity of Weak Solutions of the Navier-Stokes System, J. Differential Equations 61 (1986) 186-212. | MR 833416 | Zbl 0577.35058

[20] Hmidi T., Keraani S., Global Solutions of the Super-Critical 2D Quasi-Geostrophic Equation in Besov Spaces, Adv. Math. 214 (2) (2007) 618-638. | MR 2349714 | Zbl 1119.76070

[21] Ju N., The Maximum Principle and the Global Attractor for the Dissipative 2D Quasi-Geostrophic Equations, Commun. Math. Phys. 255 (1) (2005) 161-181. | MR 2123380 | Zbl 1088.37049

[22] Kiselev A., Nazarov F., Volberg A., Global Well-Posedness for the Critical 2D Dissipative Quasi-Geostrophic Equation, Invent. Math. 167 (3) (2007) 445-453. | MR 2276260 | Zbl 1121.35115

[23] Ladyzhenskaya O. A., On Uniqueness and Smoothness of Generalized Solutions to the Navier-Stokes Equations, Zapiski Nauchn. Seminar. POMI 5 (1967) 169-185. | MR 236541 | Zbl 0194.12805

[24] Miura H., Dissipative Quasi-Geostrophic Equation for Large Initial Data in the Critical Sobolev Space, Commun. Math. Phys. 267 (1) (2006) 141-157. | MR 2238907 | Zbl 1113.76029

[25] Pedlosky J., Geophysical Fluid Dynamics, Springer, New York, 1987. | Zbl 0713.76005

[26] Prodi G., Un Teorema Di Unicità Per El Equazioni Di Navier-Stokes, Ann. Mat. Pura Appl. 48 (1959) 173-182. | MR 126088 | Zbl 0148.08202

[27] S. Resnick, Dynamical problems in nonlinear advective partial differential equations, Ph.D. thesis, University of Chicago, 1995.

[28] Serrin J., On the Interior Regularity of Weak Solutions of the Navier-Stokes Equations, Arch. Ration. Mech. Anal. 9 (1962) 187-195. | MR 136885 | Zbl 0106.18302

[29] Wu J., Global Solutions of the 2D Dissipative Quasi-Geostrophic Equations in Besov Spaces, SIAM J. Math. Anal. 36 (3) (2004/2005) 1014-1030, (electronic). | MR 2111923 | Zbl 1083.76064

[30] Wu J., Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces, Commun. Math. Phys. 263 (3) (2006) 803-831. | MR 2211825 | Zbl 1104.35037

[31] Wu J., Existence and Uniqueness Results for the 2-D Dissipative Quasi-Geostrophic Equation, Nonlinear Anal. 67 (2007) 3013-3036. | MR 2347594 | Zbl 1122.76014