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Global well-posedness in the super-critical dissipative quasi-geostrophic equations. (English) Zbl 1019.86002
Summary: We consider the quasi-geostrophic equation with the dissipation term, $\kappa(-\Delta)^\alpha \theta$, $0\leq\alpha\leq\frac 12$. In the case $\alpha=\frac 12$, {\it P. Constantin, D. Cordoba} and {\it J. Wu} [Indiana Univ. Math. J. 50, Spec. Iss., 97-107 (2001; Zbl 0989.86004)] proved the global existence of a strong solution in $H^1$ and $H^2$ under the assumption of small $L^\infty$-norm of initial data. In this paper, we prove the global existence in the scale invariant Besov space, $\dot B^{2 - 2\alpha }_{2,1}$, $0\leq\alpha\leq\frac 12$ for initial data small in the $\dot B^{2 - 2\alpha }_{2,1}$ norm. We also prove a global stability result in $\dot B^1_{2,1}$.

86A05Hydrology, hydrography, oceanography
35Q35PDEs in connection with fluid mechanics
35A05General existence and uniqueness theorems (PDE) (MSC2000)
76D03Existence, uniqueness, and regularity theory
35B30Dependence of solutions of PDE on initial and boundary data, parameters
76U05Rotating fluids
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