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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.

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
Full Text: DOI arXiv
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