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Existence and uniqueness results for a second order differential equation for the ocean flow in arctic gyres

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Abstract

In this paper, existence and uniqueness of solutions for the nonlinear and linear models of the arctic gyres over \(84^{\circ }\) north latitude are studied by using the stereographic projection, which represents the streamline and no jet at the outside boundary. By using the fixed point technique, we prove the existence and uniqueness of the local solution of the nonlinear model. Next, we present the existence and uniqueness of the solution in the semi-infinite interval under the suitable asymptotic conditions. In the case of linear vorticity function, we give the explicit solutions by adopt the idea for linear ODEs.

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References

  1. 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 

  2. 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 

  3. Gabler, R.E., Petersen, J.F., Trapasso, L.M.: Essentials of Physical Geography. Thomson Brooks/Cole, Belmont (2007)

    Google Scholar 

  4. Garrison, T.: Essentials of Oceanography. National Geographic Society, Stamford (2014)

    Google Scholar 

  5. Apel, J.R.: Principles of Ocean Physics. Academic Press, London (1987)

    Google Scholar 

  6. 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 

  7. 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 

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

    Article  MathSciNet  Google Scholar 

  9. 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 

  10. Walton, D.W.H.: Global Science from a Frozen Continent. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  11. Wilkinson, T., Wiken, E., Bezaury-Creel, J., Hourigan, T., Agardy, T., Herrmann, H., Janishevski, L., Madden, C., Morgan, L., Padilla, M.: Marine Ecoregions of North America, p. 200. Commission for Environmental Cooperation, Montreal (2009)

    Google Scholar 

  12. Chu, J.: On a differential equation arising in geophysics. Mon. Math. 187, 499–508 (2018)

    Article  MathSciNet  Google Scholar 

  13. Chu, J.: On a nonlinear model for arctic gyres. Ann. Mat. Pura Appl. 197, 651–659 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Chu, J.: On an infinite-interval boundary-value problem in geophysics. Mon. Math. 188, 621–628 (2019)

    Article  MathSciNet  Google Scholar 

  16. Zeidler, E.: Nonlinear Functional Analysis and Its Applications, I. Fixed-Point Theorems. Springer, New York (1986). Translated from the German by Peter R. Wadsack

    Book  Google Scholar 

  17. Lipovan, O.: A retarded Gronwall-Like inequality and its applications. J. Math. Anal. Appl. 252, 389–401 (2000)

    Article  MathSciNet  Google Scholar 

  18. Miller, R.K., Michel, A.N.: Ordinary Differential Equations. Academic Press, New York (1982)

    MATH  Google Scholar 

Download references

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Authors and Affiliations

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

    Wenlin Zhang & JinRong Wang

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

    Wenlin Zhang

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

    JinRong Wang

  4. 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

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

    Michal Fečkan

Authors
  1. Wenlin Zhang
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  2. JinRong Wang
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  3. Michal Fečkan
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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], Major Research Project of Innovative Group in Guizhou Education Department [(2018)012], the Slovak Research and Development Agency under the Contract No. APVV-18-0308, the Slovak Grant Agency VEGA Nos. 1/0358/20 and 2/0127/20, and Natural Science Foundation of Guizhou Province ([2017]260)

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Zhang, W., Wang, J. & Fečkan, M. Existence and uniqueness results for a second order differential equation for the ocean flow in arctic gyres. Monatsh Math 193, 177–192 (2020). https://doi.org/10.1007/s00605-020-01388-6

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  • DOI: https://doi.org/10.1007/s00605-020-01388-6

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