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Global well-posedness for a modified critical dissipative quasi-geostrophic equation. (English) Zbl 05986533
Summary: We consider the following modified quasi-geostrophic equation $$\partial _t \delta + u \cdot \nabla \theta + \nu |D|^\alpha \theta = 0, \quad u = |D|^{\alpha -1}\cal R^\perp \theta, \quad x \in \Bbb R^2$$ with $\nu >0$ and $\alpha \in ]0,1[\cup ]1,2[$. When $\alpha \in ]0,1[$, the equation was firstly introduced by Constantin, Iyer and Wu (2008) in [11]. Here, by using the modulus of continuity method, we prove the global well-posedness of the system. As a byproduct, we also show that for every $\alpha \in ]0,2[$, the Lipschitz norm of the solution has a uniform exponential upper bound.

76U05Rotating fluids
76B03Existence, uniqueness, and regularity theory (fluid mechanics)
35Q35PDEs in connection with fluid mechanics
Full Text: DOI arXiv
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