Global in Time Classical Solutions to the 3D Quasi-geostrophic System for Large Initial Data

TL;DR

This paper proves the existence of global classical solutions for the 3D quasi-geostrophic system with large initial data, combining boundary regularization and propagation of regularity techniques.

Contribution

It establishes the first global existence result for large initial data in the 3D quasi-geostrophic system with boundary effects.

Findings

Global classical solutions exist for any smooth initial data.

The boundary regularization effect is strengthened to Besov space $ ext{B}_{ ext{infty,infty}}^1$.

The proof combines De Giorgi regularization and Beale-Kato-Majda techniques.

Abstract

In this paper, the authors show the existence of global in time classical solutions to the 3D quasi-geostrophic system with Ekman pumping for any smooth initial value (possibly large). This system couples an inviscid transport equation in $\mathbb{R}^3_+$ with an equation on the boundary satisfied by the trace. The proof combines the De Giorgi regularization effect on the boundary $z=0$ -similar to the so called surface quasi-geostrophic equation- with Beale-Kato-Majda techniques to propagate regularity for $z>0$. A potential theory argument is used to strengthen the regularization effect on the trace up to the Besov space $\mathring{B}_{\infty,\infty}^1$.