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The global attractor for the solutions to the 3D viscous primitive equations. (English) Zbl 1131.35065
The author studies the so called 3D viscous primitive equations of geophysical fluid dynamics which are given as follows: $$\aligned & \partial_t v+(v\nabla)v+w\partial_z v+\nabla p+fv^\perp+L_1v=0\\ & \partial_z p + \theta=0,\quad \nabla\cdot v+\partial_zw=0\\ & \partial_t\theta+(u\nabla)\theta+w\partial_z\theta+L_2\theta=Q.\endaligned\tag 1$$ Here $v=(v_1,v_2)$, $v^\perp=(-v_2,v_1)$, $L_i=-\nu_i\Delta - \mu_i\partial^2_z$, $i=1,2$ and $\Delta=\partial^2_x+\partial^2_y$, while $\nabla=(\partial_x,\partial_y)$. The domain underlying (1) has the form $\Omega=M\times (-h,0)$ where $M\subseteq\Bbb R^2$ is a smooth bounded domain. System (1) is supplied by appropriate (inhomogeneous) boundary conditions. Based on the third and the fourth equation in (1), system (1) can be transformed into a new, equivalent system, to be denoted by ($1'$), which involves only the unknowns $v=(v_1,v_2)$ and $\theta$ and which is supplied by homogeneous boundary conditions. It is this modified system $(1')$ which is investigated by the author. He introduces appropriate function spaces $$H_1\subseteq L^2(\Omega)^2,\quad V_1\subseteq H^1(\Omega)^2,\quad H_2=L^2(\Omega), \quad V_2= H^1(\Omega),$$ and considers $(1')$, properly interpreted, as an evolution equation on the spaces $H=H_1\times H_2$ and $V=V_1\times V_2$ respectively. A standard definition of weak solution of $(1')$ is then given, which however is not used in the sequel since a result of Cao and Titi states: given $Q\in H^1(\Omega)$ and $(v_0,\theta_0)\in V_1\times V_2$ then there is a unique global strong solution $(v,\theta)$ of $(1')$ with initial data $(v_0,\theta_0)$; the dependence on $(v_0,\theta_0)$ is continuous. By this theorem, the author can restrict attention on strong solutions and so avoid lengthy computations via the Galerkin method. The author now first proves $L^2$- and $L^6$-estimates for the temperature $\theta$, and then $L^6$- and $H^1$-estimates for the velocity vector $v=(v_1,v_2)$. The proofs of these estimates are quite difficult and based on a tricky combination of embedding inequalities and variants of Gronwalls inequality. Subsequently the continuity of strong solutions $(v(t),\theta(t))$, $t\ge 0$ with respect to time $t$ is proved. Based on this preparatory steps, the author proves his main theorem (Thm. 6.1) which asserts among others that the solution semigroup $S(t)$, $t\ge 0$, associated with the system $(1')$, admits a compact global attractor.

35Q35PDEs in connection with fluid mechanics
35B41Attractors (PDE)
86A05Hydrology, hydrography, oceanography
37L30Attractors and their dimensions, Lyapunov exponents
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