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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 $$\varphi>1$$.

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