Some Results for the Primitive Equations with Physical Boundary Conditions

TL;DR

This paper studies the 3D primitive equations with physical boundary conditions, demonstrating boundedness, decay properties, and the existence of an invariant measure under random forcing, advancing understanding of their long-term behavior.

Contribution

It establishes bounded absorbing sets, decay of solutions, and existence of invariant measures for primitive equations with physical boundary conditions, which were previously less understood.

Findings

Solutions have bounded $H^1$-norm with constant forcing.

Unforced solutions' $H^1$-norm decays to zero.

Invariant measure exists under random kick-forcing.

Abstract

In this paper we consider the (simplified) 3-dimensional primitive equations with \emph{physical boundary conditions}. We show that the equations with constant forcing have a bounded absorbing ball in the $H^1$-norm and that a solution to the unforced equations has its $H^1$-norm decay to 0. From this, we argue that there exists an invariant measure (on $H^1$) for the equations under random kick-forcing.