*(English)*Zbl 1131.35065

The author studies the so called 3D viscous primitive equations of geophysical fluid dynamics which are given as follows:

Here $v=({v}_{1},{v}_{2})$, ${v}^{\perp}=(-{v}_{2},{v}_{1})$, ${L}_{i}=-{\nu}_{i}{\Delta}-{\mu}_{i}{\partial}_{z}^{2}$, $i=1,2$ and ${\Delta}={\partial}_{x}^{2}+{\partial}_{y}^{2}$, while $\nabla =({\partial}_{x},{\partial}_{y})$. The domain underlying (1) has the form ${\Omega}=M\times (-h,0)$ where $M\subseteq {\mathbb{R}}^{2}$ 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 (${1}^{\text{'}}$), 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 $\left({1}^{\text{'}}\right)$ which is investigated by the author. He introduces appropriate function spaces

and considers $\left({1}^{\text{'}}\right)$, 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 $\left({1}^{\text{'}}\right)$ is then given, which however is not used in the sequel since a result of Cao and Titi states: given $Q\in {H}^{1}\left({\Omega}\right)$ and $({v}_{0},{\theta}_{0})\in {V}_{1}\times {V}_{2}$ then there is a unique global strong solution $(v,\theta )$ of $\left({1}^{\text{'}}\right)$ 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 $\left(v\right(t),\theta (t\left)\right)$, $t\ge 0$ 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 $S\left(t\right)$, $t\ge 0$, associated with the system $\left({1}^{\text{'}}\right)$, admits a compact global attractor.

##### MSC:

35Q35 | PDEs in connection with fluid mechanics |

35B41 | Attractors (PDE) |

86A05 | Hydrology, hydrography, oceanography |

37L30 | Attractors and their dimensions, Lyapunov exponents |