Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion

TL;DR

This paper proves the global existence and uniqueness of strong solutions for the 3D primitive equations modeling oceanic and atmospheric dynamics, even with only horizontal viscosity and diffusion, using advanced mathematical inequalities.

Contribution

It establishes the global well-posedness of strong solutions for the 3D primitive equations with minimal horizontal viscosity and diffusion, extending previous results.

Findings

Global well-posedness for initial data in H^2

Use of logarithmic Sobolev embedding inequality

Application of a system Gronwall inequality

Abstract

In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of strong solution is established for any $H^2$ initial data. An $N$-dimensional logarithmic Sobolev embedding inequality, which bounds the $L^\infty$ norm in terms of the $L^q$ norms up to a logarithm of the $L^p$-norm, for $p>N$, of the first order derivatives, and a system version of the classic Gronwall inequality are exploited to establish the required a priori $H^2$ estimates for the global regularity.