On $H^2$ solutions and $z$-weak solutions of the 3D Primitive Equations

TL;DR

This paper proves the global existence, boundedness, and uniqueness of $H^2$ and $z$-weak solutions for the 3D Primitive Equations in a bounded domain, resolving longstanding open problems in the field.

Contribution

It establishes the global well-posedness and boundedness of $H^2$ and $z$-weak solutions, improving previous results and solving an open problem for $z$-weak solutions.

Findings

Global boundedness and absorbing sets for $H^2$ solutions

Uniqueness of $z$-weak solutions

Resolution of the open problem on $z$-weak solutions

Abstract

Global in time well-posedness of $H^2$ solutions and $z$-weak solutions of the 3D Primitive equations in a bounded cylindrical domain is proved. More specifically, uniform in time boundedness and bounded absorbing sets are obtained for both $H^2$ solutions and $z$-weak solutions, as well as uniqueness of the $z$-weak solution for the 3D Primitive equations. The result for $H^2$ solutions improves a recent one proved in[6]. The result for $z$-weak solution positively resolves the problem of global existence and uniqueness of $z$-weak solutions of the 3D primitive equations, which has been open since the work of [14].