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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 $H^s(\Bbb R^2)$. 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 $s\ge 2(1-\alpha)$ 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:
35Q35PDEs in connection with fluid mechanics
86A05Hydrology, hydrography, oceanography
37L30Attractors and their dimensions, Lyapunov exponents
76D03Existence, uniqueness, and regularity theory
76D05Navier-Stokes equations (fluid dynamics)
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References:
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