The finite dimensional global attractor for the 3D viscous Primitive Equations

A new method is presented to prove finiteness of the fractal and Hausdorff dimensions of the global attractor for the strong solutions to the 3D Primitive Equations (PEs) with viscosity, which is applicable to more general situations than the recent result of [8] in the sense that it removes all extra technical conditions imposed by previous analyses. More specifically, the dimensions of the global attractor are proved finite for heat source $Q\in L^2$, exactly the same condition for well-posedness of global strong solutions and existence of the global attractor of these solutions; while the best previous result obtained recently in [8] still requires the extra condition that $∂_zQ\in L^2$ for finiteness of the dimensions of the global attractor. The key new idea is that Ladyzhenskaya's squeezing property of the semigroup for the strong solutions can be established without higher solution regularity of Primitive Equations. This has the general interest for dissipative evolution equations. For this reason, the new method especially has the advantange of dealing with more complicated boundary conditions which present essential difficulties for previous methods. In particular, the case of 3D viscous PEs with `` physical boundary conditions'' can be treated by the new method in the same way as presented in this article, however, it seems rather difficult for previous methods.