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Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. (English) Zbl 1106.35061

Summary: We study the two-dimensional dissipative quasi-geostrophic equations

${\theta }_{t}+u·\nabla \theta =0,\phantom{\rule{2.em}{0ex}}{\theta }_{t}+u·\nabla \theta +\kappa {\left(-{\Delta }\right)}^{\alpha }\theta =0,$
$u=\left({u}_{1},{u}_{2}\right)=\left(-\frac{\partial \psi }{\partial {x}_{2}},\frac{\partial \psi }{\partial {x}_{1}}\right),\phantom{\rule{2.em}{0ex}}{\left(-{\Delta }\right)}^{1/2}\psi =-\theta ,$

in the Sobolev space ${H}^{s}\left({ℝ}^{2}\right)$. Existence and uniqueness of the solution local in time is proved in ${H}^{s}$ when $s>2\left(1-\alpha \right)$. Existence and uniqueness of the solution global in time is also proved in ${H}^{s}$ when $s\ge 2\left(1-\alpha \right)$ and the initial data $\parallel {{\Lambda }}^{s}{\theta }_{0}{\parallel }_{{L}^{2}}$ is small. For the case, $s>2\left(1-\alpha \right)$, we also obtain the unique large global solution in ${H}^{s}$ provided that $\parallel {\theta }_{0}{\parallel }_{{L}^{2}}$ is small enough.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 86A05 Hydrology, hydrography, oceanography 37L30 Attractors and their dimensions, Lyapunov exponents 76D03 Existence, uniqueness, and regularity theory 76D05 Navier-Stokes equations (fluid dynamics)
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