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.
\[ \theta_t+u\cdot \nabla\theta= 0, \qquad \theta_t+u\cdot \nabla\theta+ \kappa(-\Delta)^\alpha\theta= 0, \]
\[ u=(u_1,u_2)= \biggl(- \frac{\partial\psi} {\partial x_2}, \frac{\partial\psi} {\partial x_1} \biggr), \qquad (-\Delta)^{1/2}\psi=-\theta, \]
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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