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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
$\theta_t+u\cdot \nabla\theta= 0, \qquad \theta_t+u\cdot \nabla\theta+ \kappa(-\Delta)^\alpha\theta= 0,$
$u=(u_1,u_2)= \biggl(- \frac{\partial\psi} {\partial x_2}, \frac{\partial\psi} {\partial x_1} \biggr), \qquad (-\Delta)^{1/2}\psi=-\theta,$
in the Sobolev space $$H^s(\mathbb R^2)$$. Existence and uniqueness of the solution local in time is proved in $$H^s$$ when $$s>2(1-\alpha)$$. Existence and uniqueness of the solution global in time is also proved in $$H^s$$ when $$s\geq 2(1-\alpha)$$ and the initial data $$\|\Lambda^s\theta_0 \|_{L^2}$$ is small. For the case, $$s>2(1-\alpha)$$, we also obtain the unique large global solution in $$H^s$$ provided that $$\|\theta_0\|_{L^2}$$ is small enough.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 86A05 Hydrology, hydrography, oceanography 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids
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