Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere

TL;DR

This paper proves the finite dimensionality of the global attractor for solutions to 3D moist primitive equations of the atmosphere by establishing uniform bounds and continuity properties despite boundary and structural complexities.

Contribution

It introduces a novel approach using time derivative bounds to demonstrate the attractor's finite dimensionality under complex boundary conditions.

Findings

Proved uniform bounds for strong solutions in $L^2( abla)$

Established the global attractor's uniform continuity

Demonstrated the finiteness of Hausdorff and fractal dimensions

Abstract

Under general boundary conditions we consider the finiteness of the Hausdorff and fractal dimensions of the global attractor for the strong solution of the 3D moist primitive equations with viscosity. Firstly, we obtain time-uniform estimates of the first-order time derivative of the strong solutions in $L^2(\mho)$. Then, to prove the finiteness of the Hausdorff and fractal dimensions of the global attractor, the common method is to obtain the uniform boundedness of the strong solution in $H^2(\mho)$ to establish the squeezing property of the solution operator. But it is difficult to achieve due to the boundary conditions and complicated structure of the 3D moist primitive equations. To overcome the difficulties, we try to use the uniform boundedness of the derivative of the strong solutions with respect to time $t$ in $L^2(\mho)$ to prove the uniform continuity of the global attractor. Finally, using the uniform continuity of the global attractor we establish the squeezing property of the solution operator which implies the finiteness of the Hausdorff and fractal dimensions of the global attractor.