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Global well-posedness for a modified critical dissipative quasi-geostrophic equation. (English) Zbl 1382.35233
Summary: We consider the following modified quasi-geostrophic equation $\partial _t \delta + u \cdot \nabla \theta + \nu |D|^\alpha \theta = 0, \quad u = |D|^{\alpha -1}\mathcal R^\perp \theta, \quad x \in \mathbb R^2$ with $$\nu >0$$ and $$\alpha \in ]0,1[\cup ]1,2[$$. When $$\alpha \in ]0,1[$$, the equation was firstly introduced by P. Constantin et al. [Indiana Univ. Math. J. 57, No. 6, 2681–2692 (2008; Zbl 1159.35059)]. 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.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B44 Blow-up in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 76U05 General theory of rotating fluids
Zbl 1159.35059
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