Time Periodic Doubly Connected Solutions for the 3D Quasi-Geostrophic Model

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 ≥ m0 and m0 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.