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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 $$\alpha=\frac 12$$, P. Constantin, D. Cordoba and 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 $$\dot B^1_{2,1}$$.

MSC:
 86A05 Hydrology, hydrography, oceanography 35Q35 PDEs in connection with fluid mechanics 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 76U05 General theory of rotating fluids
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