# zbMATH — the first resource for mathematics

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. (English) Zbl 1149.76052
Summary: We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical $$(\alpha <1/2)$$ dissipation $$( - \Delta )^{\alpha}$$: If a Leray-Hopf weak solution is Hölder continuous $$\theta \in C^{\delta }(\mathbb R^2)$$ with $$\delta >1 - 2\alpha$$ on the time interval $$[t_{0},t]$$, then it is actually a classical solution on ($$t_{0},t$$].

##### MSC:
 76U05 General theory of rotating fluids 35Q35 PDEs in connection with fluid mechanics 86A10 Meteorology and atmospheric physics 86A05 Hydrology, hydrography, oceanography
##### Keywords:
supercritical dissipation; Leray-Hopf weak solution
Full Text:
##### References:
 [1] Caffarelli, L.; Vasseur, A., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, (2006), arXiv: [2] Chae, D., On the regularity conditions for the dissipative quasi-geostrophic equations, SIAM J. math. anal., 37, 1649-1656, (2006) · Zbl 1141.76010 [3] Chae, D.; Lee, J., Global well-posedness in the super-critical dissipative quasi-geostrophic equations, Commun. math. phys., 233, 297-311, (2003) · Zbl 1019.86002 [4] Chen, Q.; Miao, C.; Zhang, Z., A new Bernstein inequality and the 2D dissipative quasi-geostrophic equation, Commun. math. phys., 271, 821-838, (2007) · Zbl 1142.35069 [5] Constantin, P., Euler equations, navier – stokes equations and turbulence, (), 1-43 · Zbl 1190.76146 [6] Constantin, P.; Cordoba, D.; Wu, J., On the critical dissipative quasi-geostrophic equation, Indiana univ. math. J., 50, 97-107, (2001) · Zbl 0989.86004 [7] Constantin, P.; Majda, A.; Tabak, E., Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar, Nonlinearity, 7, 1495-1533, (1994) · Zbl 0809.35057 [8] Constantin, P.; Wu, J., Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. math. anal., 30, 937-948, (1999) · Zbl 0957.76093 [9] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Commun. math. phys., 249, 511-528, (2004) · Zbl 1309.76026 [10] Held, I.; Pierrehumbert, R.; Garner, S.; Swanson, K., Surface quasi-geostrophic dynamics, J. fluid mech., 282, 1-20, (1995) · Zbl 0832.76012 [11] T. Hmidi, S. Keraani, On the global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces, Adv. Math., in press · Zbl 1119.76070 [12] Ju, N., The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Commun. math. phys., 255, 161-181, (2005) · Zbl 1088.37049 [13] Ju, N., Global solutions to the two dimensional quasi-geostrophic equation with critical or super-critical dissipation, Math. ann., 334, 627-642, (2006) · Zbl 1145.76053 [14] Kiselev, A.; Nazarov, F.; Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. math., 167, 445-453, (2007) · Zbl 1121.35115 [15] Marchand, F.; Lemarié-Rieusset, P.G., Solutions auto-similaires non radiales pour l’équation quasi-géostrophique dissipative critique, C. R. math. acad. sci. Paris, ser. I, 341, 535-538, (2005) · Zbl 1155.76320 [16] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer New York · Zbl 0713.76005 [17] S. Resnick, Dynamical problems in nonlinear advective partial differential equations, Ph.D. thesis, University of Chicago, 1995 [18] Schonbek, M.; Schonbek, T., Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. math. anal., 35, 357-375, (2003) · Zbl 1126.76386 [19] Schonbek, M.; Schonbek, T., Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows, Discrete contin. dyn. syst., 13, 1277-1304, (2005) · Zbl 1091.35070 [20] Wu, J., The quasi-geostrophic equation and its two regularizations, Comm. partial differential equations, 27, 1161-1181, (2002) · Zbl 1012.35067 [21] Wu, J., Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces, SIAM J. math. anal., 36, 1014-1030, (2004/2005) · Zbl 1083.76064 [22] Wu, J., The quasi-geostrophic equation with critical or supercritical dissipation, Nonlinearity, 18, 139-154, (2005) · Zbl 1067.35002 [23] Wu, J., Solutions of the 2-D quasi-geostrophic equation in Hölder spaces, Nonlinear anal., 62, 579-594, (2005) · Zbl 1116.35348 [24] Wu, J., Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation, Nonlinear anal., 67, 3013-3036, (2007) · Zbl 1122.76014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.