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A frequency localized maximum principle applied to the 2D quasi-geostrophic equation. (English) Zbl 1248.35211
Summary: In this paper, we prove a maximum principle for a frequency localized transport-diffusion equation. As an application, we prove the local well-posedness of the supercritical quasi-geostrophic equation in the critical Besov spaces $$\overset \circ {B}^{1-\alpha}_{\infty,q}$$, and global well-posedness of the critical quasi-geostrophic equation in $${\overset \circ {B}^{0}_{\infty,q}}$$ for all $$1 \leq q < \infty$$. Here $${\overset \circ {B}^{s}_{\infty,q} }$$ is the closure of the Schwartz functions in the norm of $${B^{s}_{\infty,q}}$$.

##### MSC:
 35Q86 PDEs in connection with geophysics 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 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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