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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:

$\begin{array}{cc}& {\partial }_{t}v+\left(v\nabla \right)v+w{\partial }_{z}v+\nabla p+f{v}^{\perp }+{L}_{1}v=0\hfill \\ & {\partial }_{z}p+\theta =0,\phantom{\rule{1.em}{0ex}}\nabla ·v+{\partial }_{z}w=0\hfill \\ & {\partial }_{t}\theta +\left(u\nabla \right)\theta +w{\partial }_{z}\theta +{L}_{2}\theta =Q·\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

Here $v=\left({v}_{1},{v}_{2}\right)$, ${v}^{\perp }=\left(-{v}_{2},{v}_{1}\right)$, ${L}_{i}=-{\nu }_{i}{\Delta }-{\mu }_{i}{\partial }_{z}^{2}$, $i=1,2$ and ${\Delta }={\partial }_{x}^{2}+{\partial }_{y}^{2}$, while $\nabla =\left({\partial }_{x},{\partial }_{y}\right)$. The domain underlying (1) has the form ${\Omega }=M×\left(-h,0\right)$ where $M\subseteq {ℝ}^{2}$ 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 $v=\left({v}_{1},{v}_{2}\right)$ 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

${H}_{1}\subseteq {L}^{2}{\left({\Omega }\right)}^{2},\phantom{\rule{1.em}{0ex}}{V}_{1}\subseteq {H}^{1}{\left({\Omega }\right)}^{2},\phantom{\rule{1.em}{0ex}}{H}_{2}={L}^{2}\left({\Omega }\right),\phantom{\rule{1.em}{0ex}}{V}_{2}={H}^{1}\left({\Omega }\right),$

and considers $\left({1}^{\text{'}}\right)$, properly interpreted, as an evolution equation on the spaces $H={H}_{1}×{H}_{2}$ and $V={V}_{1}×{V}_{2}$ 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 $\left({v}_{0},{\theta }_{0}\right)\in {V}_{1}×{V}_{2}$ then there is a unique global strong solution $\left(v,\theta \right)$ of $\left({1}^{\text{'}}\right)$ with initial data $\left({v}_{0},{\theta }_{0}\right)$; the dependence on $\left({v}_{0},{\theta }_{0}\right)$ 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 $v=\left({v}_{1},{v}_{2}\right)$. 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 $\left(v\left(t\right),\theta \left(t\right)\right)$, $t\ge 0$ 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