Existence and regularity results for the inviscid primitive equations with lateral periodicity

TL;DR

This paper proves the existence and regularity of solutions for the 3D inviscid linearized primitive equations in a channel with lateral periodicity, addressing a gap in the mathematical understanding of these hyperbolic systems.

Contribution

It establishes well-posedness and regularity results for the inviscid primitive equations with non-local boundary conditions, extending previous work on viscous boundary layers.

Findings

Proved existence of solutions for the inviscid LPEs.

Established regularity properties of these solutions.

Introduced non-local boundary conditions for well-posedness.

Abstract

The article is devoted to prove the existence and regularity of the solutions of the $3D$ inviscid Linearized Primitive Equations (LPEs) in a channel with lateral periodicity. This was assumed in a previous work \cite{HJT} which is concerned with the boundary layers generated by the corresponding viscous problem. Although the equations under investigation here are of hyperbolic type, the standard methods do not apply because of the specificity of the hyperbolic system. A set of \textit{non-local} boundary conditions for the inviscid LPEs has to be imposed at the top and bottom of the channel making thus the system well-posed.