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Solutions of the 2D quasi-geostrophic equation in Hölder spaces. (English) Zbl 1116.35348
Summary: The 2D quasi-geostrophic equation \[ \partial_t\theta+u\cdot\nabla \theta+\kappa(-\Delta)^\alpha\theta=0,\quad u={\mathcal R}^\perp (\theta) \] is a two-dimensional model of the 3D hydrodynamics equations. When \(\alpha\leq\frac 12\), the issue of existence and uniqueness concerning this equation becomes difficult. It is shown here that this equation with either \(\square=0\) or \(\square>0\) and \(0\leq \alpha \leq\frac 12\) has a unique local in time solution corresponding to any initial datum in the space \(C^rL^q\) for \(r<1\) and \(\varphi>1\).

35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76U05 General theory of rotating fluids
Full Text: DOI
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