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Positive solutions to integral boundary value problems from geophysical fluid flows

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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.

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References

  1. Constantin, A., Johnson, R.S.: Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates. Proc. R. Soc. A Math. 473, 20170063 (2017)

    Article  MathSciNet  Google Scholar 

  2. Viudez, A., Dritschel, D.G.: Vertical velocity in mesoscale geophysical flows. J. Fluid Mech. 483, 199–223 (2015)

    Article  MathSciNet  Google Scholar 

  3. Talley, L.D., Pickard, G.L., Emery, W.J., Swift, J.H.: Descriptive Physical Oceanography: An Introduction. Academic Press, New York (2011)

    Book  Google Scholar 

  4. Danabasoglu, G., McWilliams, J.C., Gent, P.R.: The role of mesoscale tracer transport in the global ocean circulation. Science 264, 1123–1126 (1994)

    Article  Google Scholar 

  5. Farneti, R., Delworth, T.L., Rosati, A., Griffies, S.M., Zeng, F.: The role of mesoscale eddies in the rectification of the Southern Ocean response to climate change. J. Phys. Oceanogr. 40, 1539–1557 (2010)

    Article  Google Scholar 

  6. Firing, Y.L., Chereskin, T.K., Mazloff, M.R.: Vertical structure and transport of the Antarctic circumpolar current in drake passage from direct velocity observations. J. Geophys. Res. Atmos. 116, C08015 (2011)

    Article  Google Scholar 

  7. Ivchenko, V.O., Richards, K.J.: The dynamics of the Antarctic circumpolar current. J. Phys. Oceanogr. 26, 753–774 (2012)

    Article  Google Scholar 

  8. Klinck, J., Nowlin, W.D.: Antarctic Circumpolar Current. Academic Press, Cambridge (2001)

    Book  Google Scholar 

  9. Marshall, D.P., Munday, D.R., Allison, L.C., Hay, R.J., Johnson, H.L.: Gill’s model of the Antarctic circumpolar current, revisited: the role of latitudinal variations in wind stress. Ocean Model. 97, 37–51 (2016)

    Article  Google Scholar 

  10. Tomczak, M., Godfrey, J.S.: Regional Oceanography: An Introdution. Pergamon Press, Oxford (1994)

    Google Scholar 

  11. Thompson, A.F.: The atmospheric ocean: eddies and jets in the Antarctic circumpolar current. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 366, 4529–4541 (2008)

    Article  MathSciNet  Google Scholar 

  12. White, W.B., Peterson, R.G.: An antarctic circumpolar wave in surface pressure, wind, temperature and sea-ice extent. Nature 380, 699–702 (1996)

    Article  Google Scholar 

  13. Wolff, J.O.: Modelling the antarctic circumpolar current: eddy-dynamics and their parametrization. Environ. Model Softw. 14, 317–326 (1999)

    Article  Google Scholar 

  14. Constantin, A., Johnson, R.S.: An exact, steady, purely azimuthal flow as a model for the Antarctic circumpolar current. J. Phys. Oceanogr. 46, 3585–3594 (2016)

    Article  Google Scholar 

  15. Marynets, K.: A nonlinear two-point boundary-value problem in geophysics. Monatshefte für Mathematik 188, 287–295 (2019)

    Article  MathSciNet  Google Scholar 

  16. Marynets, K.: On a two-point boundary-value problem in geophysics. Appl. Anal. 98, 553–560 (2019)

    Article  MathSciNet  Google Scholar 

  17. Chu, J.: On a differential equation arising in geophysics. Monatshefte für Mathematik 187, 499–508 (2018)

    Article  MathSciNet  Google Scholar 

  18. Chu, J.: On a nonlinear model for arctic gyres. Annali di Matematica 197, 651–659 (2018)

    Article  MathSciNet  Google Scholar 

  19. Chu, J.: Monotone solutions of a nonlinear differential equation for geophysical fluid flows. Nonlinear Anal. 166, 144–153 (2018)

    Article  MathSciNet  Google Scholar 

  20. Chu, J.: On an infinite-interval boundary-value problem in geophysics. Monatshefte für Mathematik 188, 621–628 (2019)

    Article  MathSciNet  Google Scholar 

  21. Marynets, K.: A weighted Sturm–Liouville problem related to ocean flows. J. Math. Fluid Mech. 20, 929–935 (2018)

    Article  MathSciNet  Google Scholar 

  22. Zhang, W., Wang, J., Fečkan, M.: Existence and uniqueness results for a second order differential equation for the ocean flow in arctic gyres. Monatshefte für Mathematik 193, 177–192 (2020)

    Article  MathSciNet  Google Scholar 

  23. Calin, L.M., Ronald, Q.: A steady stratified purely azimuthal flow representing the Antarctic circumpolar current. Monatshefte für Mathematik 192, 401–407 (2020)

    Article  MathSciNet  Google Scholar 

  24. Calin, L.M., Ronald, Q.: Explicit and exact solutions concerning the Antarctic circumpolar current with variable density in spherical coordinate. J. Math. Phys. 60, 101505 (2019)

    Article  MathSciNet  Google Scholar 

  25. Zhang, X.G., Jiang, J.Q., Wu, Y.H., Cui, Y.J.: Existence and asymptotic properties of solutions for a nonlinear Schrödinger elliptic equation from geophysical fluid flows. Appl. Math. Lett. 90, 229–237 (2019)

    Article  MathSciNet  Google Scholar 

  26. Constantin, A.: Nonlinear water waves with applications to wave-current interactions and tsunamis. In: CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia (2011)

  27. Constantin, A., Strauss, W., Varvaruca, E.: Global bifurcation of steady gravity water waves with critical layers. Acta Math. 217, 195–262 (2016)

    Article  MathSciNet  Google Scholar 

  28. Ewing, J.A.: Wind, wave and current data for the design of ships and offshore structures. Mar. Struct. 3, 421–459 (1990)

    Article  Google Scholar 

  29. Constantin, A., Johnson, R.S.: The dynamics of waves interacting with the equatorial undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358 (2015)

    Article  MathSciNet  Google Scholar 

  30. Constantin, A., Johnson, R.S.: An exact, steady, purely azimuthal equatorial flow with a free surface. J. Phys. Oceanogr. 46, 1935–1945 (2016)

    Article  Google Scholar 

  31. Constantin, A., Monismith, S.G.: Gerstner waves in the presence of mean currents and rotation. J. Fluid Mech. 820, 511–528 (2017)

    Article  MathSciNet  Google Scholar 

  32. Henry, D.: Large amplitude steady periodic waves for fixed-depth rotational flows. Commun. Partial Differ. Equ. 38, 1015–1037 (2013)

    Article  MathSciNet  Google Scholar 

  33. Thomas, G.P.: Wave-current interactions: an experimental and numerical study. J. Fluid Mech. 216, 505–536 (1990)

    Article  Google Scholar 

  34. Zhang, Z.T., Wang, K.L.: On fixed point theorems of mixed monotone operators and applications. Nonlinear Anal. 70, 3279–3284 (2009)

    Article  MathSciNet  Google Scholar 

  35. Kong, L.J.: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal. 72, 2628–2638 (2010)

    Article  MathSciNet  Google Scholar 

  36. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cone. Academic Press, New York (1988)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, China

    Wenlin Zhang & JinRong Wang

  2. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská Dolina, 842 48, Bratislava, Slovakia

    Michal Fečkan

  3. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Michal Fečkan

  4. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, China

    JinRong Wang

  5. School of Mathematics and Computer Science, Liupanshui Normal University, Liupanshui, 553004, Guizhou, China

    Wenlin Zhang

Authors
  1. Wenlin Zhang
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  2. Michal Fečkan
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  3. JinRong Wang
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Corresponding author

Correspondence to JinRong Wang.

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Communicated by Adrian Constantin.

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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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Zhang, W., Fečkan, M. & Wang, J. Positive solutions to integral boundary value problems from geophysical fluid flows. Monatsh Math 193, 901–925 (2020). https://doi.org/10.1007/s00605-020-01467-8

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  • DOI: https://doi.org/10.1007/s00605-020-01467-8

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