Geophysics and Stuart vortices on a sphere meet differential geometry

Łukasz Rudnicki

doi:10.3934/cpaa.2022075

Article Contents

2022, Volume 21, Issue 7: 2479-2493. Doi: 10.3934/cpaa.2022075

This issue Previous Article The surface current of Ekman flows with time-dependent eddy viscosity Next Article

Geophysics and Stuart vortices on a sphere meet differential geometry

Łukasz Rudnicki

International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland

Received: December 2021

Revised: March 2022

Early access: April 2022

Published: July 2022

Łukasz Rudnicki is supported by the Foundation for Polish Science (IRAP project, ICTQT, Contract No. 2018/MAB/5, cofinanced by the EU within the Smart Growth Operational Programme)

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Abstract

We prove new existence criteria relevant for the non-linear elliptic PDE of the form $ \Delta_{S^2} u = C-he^{u} $, considered on a two dimensional sphere $ S^2 $, in the parameter regime $ 2\leq C<4 $. We apply this result, as well as several previously known results valid when $ C<2 $, to discuss existence of solutions of a particular PDE modelling ocean surface currents.

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Mathematics Subject Classification: Primary: 86A10, 35A01; Secondary: 58J05, 35Q35, 35J15, 35J60.

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