Finite Dimensionality of the Global Attractor for the Solutions to 3D Primitive Equations with Viscosity

TL;DR

This paper introduces a new method to prove the finite dimensionality of the global attractor for 3D Primitive Equations with viscosity, removing previous technical restrictions and broadening applicability.

Contribution

The paper presents a novel approach that establishes finiteness of the global attractor's dimensions under minimal conditions, applicable to more general boundary conditions.

Findings

Finiteness of the global attractor's fractal and Hausdorff dimensions is proven.

The method requires only the heat source in L^2, matching the existence condition for strong solutions.

The approach applies to complex boundary conditions previously inaccessible.

Abstract

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 with viscosity, which is applicable to even more general situations than the recent result of [7] in the sense that it removes all extra technical conditions imposed by previous analyses. More specifically, for finiteness of the dimensions of the global attractor, we only need the heat source $Q\in L^2$ which is exactly the condition for the existence of global strong solutions and the existence of the global attractor of these solutions; while the best existing result, which was obtained very recently in [7], still needs the extra condition that $\partial_z Q\in L^2 $ for finiteness of the dimensions of the global attractor. Moreover, the new method can be applied to cases with more complicated boundary conditions which present essential difficulties for previous methods.