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August  2019, 39(8): 4771-4781. doi: 10.3934/dcds.2019194

## Study of a nonlinear boundary-value problem of geophysical relevance

 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

* Corresponding author: Kateryna Marynets

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received  October 2018 Revised  December 2018 Published  May 2019

Fund Project: The author is supported by the WWTF research grant MA16-009

We present some results on the existence and uniqueness of solutions of a two-point nonlinear boundary value problem that arises in the modeling of the flow of the Antarctic Circumpolar Current.

Citation: Kateryna Marynets. Study of a nonlinear boundary-value problem of geophysical relevance. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4771-4781. doi: 10.3934/dcds.2019194
##### References:
 [1] A. Constantin, Global existence of solutions for perturbed differential equations, Ann. Mat. Pura Appl., 168 (1995), 237-299. doi: 10.1007/BF01759263. [2] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, 81, SIAM, Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873. [3] A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current, J. Phys. Oceanogr., 46 (2016), 3585-3594. doi: 10.1175/JPO-D-16-0121.1. [4] A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. Roy. Soc. London A, 473 (2017), 20170063, 17 pp. doi: 10.1098/rspa.2017.0063. [5] A. Constantin and R. S. Johnson, Steady large-scale ocean flows in spherical coordinates, Oceanography, 31 (2018), 42-50. doi: 10.5670/oceanog.2018.308. [6] A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528. doi: 10.1017/jfm.2017.223. [7] A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers, Acta Mathematica, 217 (2016), 195-262. doi: 10.1007/s11511-017-0144-x. [8] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, Mass. 1965. [9] J. A. Ewing, Wind, wave and current data for the design of ships and offshore structures, Marine Structures, 3 (1990), 421-459. doi: 10.1016/0951-8339(90)90001-8. [10] S. V. Haziot and K. Marynets, Applying the stereographic projection to the modeling of the flow of the Antarctic Circumpolar Current, Oceanography, 31 (2018), 68-75. [11] H.-C. Hsu and C. I. Martin, On the existence of solutions and the pressure function related to the Antarctic Circumpolar Current, Nonlinear Anal., 155 (2017), 285-293. doi: 10.1016/j.na.2017.02.021. [12] D. Ionescu-Kruse, Local stability for an exact steady purely azimuthal flow which models the Antarctic Circumpolar Current, J. Math. Fluid Mech., 20 (2018), 569-579. doi: 10.1007/s00021-017-0335-4. [13] K. Marynets, On a two-point boundary-value problem in geophysics, Applicable Analysis, 98 (2019), 553-560. doi: 10.1080/00036811.2017.1395869. [14] K. Marynets, A nonlinear two-point boundary-value problem in geophysics, Monatsh Math., 188 (2019), 287-295. doi: 10.1007/s00605-017-1127-x. [15] K. Marynets, Two-point boundary-value problem for modeling the jet flow of the Antarctic Circumpolar Current, Electronic J. Diff. Eq., 56 (2018), Paper No. 56, 12 pp. [16] O. G. Mustafa and Y. V. Rogovchenko, Global existence of solutions for a class of nonlinear differential equations, Appl. Math. Letters, 16 (2003), 753-758. doi: 10.1016/S0893-9659(03)00078-8. [17] R. Quirchmayr, A steady, purely azimuthal flow model for the Antarctic Circumpolar Current, Monatsh. Math., 187 (2018), 565-572. doi: 10.1007/s00605-017-1097-z. [18] K. Schrader and P. Waltman, An existence theorem for nonlinear boundary value problems, Proc. Amer. Math. Soc., 21 (1969), 653-656. doi: 10.1090/S0002-9939-1969-0239176-0. [19] D. W. H. Walton, Antarctica: Global Science from a Frozen Continent, Cambridge University Press, Cambridge, 2013. doi: 10.1017/CBO9780511782299.

show all references

##### References:
 [1] A. Constantin, Global existence of solutions for perturbed differential equations, Ann. Mat. Pura Appl., 168 (1995), 237-299. doi: 10.1007/BF01759263. [2] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, 81, SIAM, Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873. [3] A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current, J. Phys. Oceanogr., 46 (2016), 3585-3594. doi: 10.1175/JPO-D-16-0121.1. [4] A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. Roy. Soc. London A, 473 (2017), 20170063, 17 pp. doi: 10.1098/rspa.2017.0063. [5] A. Constantin and R. S. Johnson, Steady large-scale ocean flows in spherical coordinates, Oceanography, 31 (2018), 42-50. doi: 10.5670/oceanog.2018.308. [6] A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528. doi: 10.1017/jfm.2017.223. [7] A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers, Acta Mathematica, 217 (2016), 195-262. doi: 10.1007/s11511-017-0144-x. [8] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, Mass. 1965. [9] J. A. Ewing, Wind, wave and current data for the design of ships and offshore structures, Marine Structures, 3 (1990), 421-459. doi: 10.1016/0951-8339(90)90001-8. [10] S. V. Haziot and K. Marynets, Applying the stereographic projection to the modeling of the flow of the Antarctic Circumpolar Current, Oceanography, 31 (2018), 68-75. [11] H.-C. Hsu and C. I. Martin, On the existence of solutions and the pressure function related to the Antarctic Circumpolar Current, Nonlinear Anal., 155 (2017), 285-293. doi: 10.1016/j.na.2017.02.021. [12] D. Ionescu-Kruse, Local stability for an exact steady purely azimuthal flow which models the Antarctic Circumpolar Current, J. Math. Fluid Mech., 20 (2018), 569-579. doi: 10.1007/s00021-017-0335-4. [13] K. Marynets, On a two-point boundary-value problem in geophysics, Applicable Analysis, 98 (2019), 553-560. doi: 10.1080/00036811.2017.1395869. [14] K. Marynets, A nonlinear two-point boundary-value problem in geophysics, Monatsh Math., 188 (2019), 287-295. doi: 10.1007/s00605-017-1127-x. [15] K. Marynets, Two-point boundary-value problem for modeling the jet flow of the Antarctic Circumpolar Current, Electronic J. Diff. Eq., 56 (2018), Paper No. 56, 12 pp. [16] O. G. Mustafa and Y. V. Rogovchenko, Global existence of solutions for a class of nonlinear differential equations, Appl. Math. Letters, 16 (2003), 753-758. doi: 10.1016/S0893-9659(03)00078-8. [17] R. Quirchmayr, A steady, purely azimuthal flow model for the Antarctic Circumpolar Current, Monatsh. Math., 187 (2018), 565-572. doi: 10.1007/s00605-017-1097-z. [18] K. Schrader and P. Waltman, An existence theorem for nonlinear boundary value problems, Proc. Amer. Math. Soc., 21 (1969), 653-656. doi: 10.1090/S0002-9939-1969-0239176-0. [19] D. W. H. Walton, Antarctica: Global Science from a Frozen Continent, Cambridge University Press, Cambridge, 2013. doi: 10.1017/CBO9780511782299.
Depiction of the azimuthal and polar spherical coordinates $\varphi \in [0,2\pi)$ and $\theta \in [0,\pi]$ of a point $P$ on the spherical surface of the Earth: $\theta = 0$ corresponds to the North Pole and $\theta = \pi/2$ to the Equator
The unit vectors of the coordinate system on the eastward rotating spherical Earth, with $e_{\varphi}$ pointing in the azimuthal direction
Depiction of the stereographic projection $P \mapsto P'$ from the North Pole to the equatorial plane, illustrated for a location that corresponds to the region where the Antarctic Circumplolar Current is encountered
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