Time Periodic Doubly Connected Solutions for the 3D Quasi-Geostrophic Model
Résumé
In this paper, we construct time periodic doubly connected solutions for the three-dimensional 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 >= m(0) and m(0) large enough. The linearized matrix operator at the equilibrium state has variable and singular coefficients and its spectral analysis is performed via the approach devised in [C. Garc & iacute;a, T. Hmidi, and J. Mateu, Comm. Math. Phys., 390 (2022), pp. 617-756], where a suitable symmetrization was introduced. New difficulties emerge due to the interaction between the surfaces making the spectral problem richer and involved.