Time periodic doubly connected solutions for the 3D quasi-geostrophic model

TL;DR

This paper constructs time-periodic, doubly connected rotating solutions for the 3D quasi-geostrophic model, revealing complex spectral interactions and bifurcations from symmetric shapes in a mathematically rigorous framework.

Contribution

It introduces a novel method for proving the existence of doubly connected rotating patches in 3D quasi-geostrophic models, including spectral analysis of variable coefficient operators.

Findings

Existence of nontrivial doubly connected rotating patches.

Spectral analysis of linearized operators with variable, singular coefficients.

Bifurcation from symmetric shape domains with high symmetry.

Abstract

In this paper, we construct time periodic doubly connected solutions for the 3D quasi-geostrophic model in the patch setting. More specifically, we prove the existence of nontrivial $m$-fold doubly connected rotating patches bifurcating from a generic doubly connected revolution shape domain with higher symmetry $m\geq m_0$ and $m_0$ is large enough. The linearized matrix operator at the equilibrium state is with variable and singular coefficients and its spectral analysis is performed via the approach devised in [27] where a suitable symmetrization has been introduced. New difficulties emerge due to the interaction between the surfaces making the spectral problem richer and involved.