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ositive solutions to integral boundary value problems from geophysical fluid flows Wenlin Zhang1,5 · Michal Feˇckan2,3 · JinRong Wang1,4 Received: 2 June 2020 / Accepted: 15 September 2020 / Published online: 22 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020

Abstract The mathematical model of the Antarctic Circumpolar Current with integral boundary conditions is established and the explicit expression of green’s function is obtained. The existence and uniqueness of solutions are proved by using the mixed monotone operator theory. The sufficient conditions for the existence of positive solutions of the model are given and the existence of positive solutions with integral boundary is proved by using the fixed point technique in cone. Keywords Antarctic circumpolar current · Positive solutions · Fixed point theory Mathematics Subject Classification 34A12 · 45G99 · 76B03

1 Introduction The mathematical study of ocean circulation is very important for predicting the characteristics of large-scale natural phenomena in the ocean. The combined forces of gravity and Coriolis forces (due to the earth’s rotation), triggered by wind stress, drive circulating ocean currents, known as gyres. In the gyres, the horizontal velocity is 0.01 m/s, which is about 104 times the vertical velocity in [1,2]. Considering global ocean circulation and global climate, the Antarctic Circumpolar Current (ACC) is probably the most important current of this type. ACC is the most powerful ocean current on

Communicated by Adrian Constantin. This work is partially supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Natural Science Foundation of Guizhou Province ([2018]387), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), the Slovak Research and Development Agency under the contract No. APVV-18-0308, and the Slovak Grant Agency VEGA Nos. 1/0358/20 and 2/0127/20.

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JinRong Wang [email protected]

Extended author information available on the last page of the article

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earth in [3], which separates Antarctica from the warm subtropical waters. ACC flows clockwise from west to east around the South Pole between about 40◦ S and 60◦ S, where there is no large area of land to break down this continuous water. ACC is the only tidal current that completely circles the earth, and its eastward flow is caused by a combination of very strong westerly winds and Coriolis forces. ACC is strongly constrained by the terrain at the bottom, and observed time changes, such as the Antarctic circumpolar wave [4–13]. ACC carries about 140 million cubic meters of water per second, more than 100 times more than all the world’s rivers combined, and travels about 24,000 km. Nevertheless, ACC is one of the least representative components of global climate models in [14]. Although there are a lot of observations about ACC flow, the pursuit of models that show a high degree of real