• Previous Article
    On stratified water waves with critical layers and Coriolis forces
  • DCDS Home
  • This Issue
  • Next Article
    On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density
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. ConstantinW. 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. ConstantinW. 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.
Figure 1.  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
Figure 2.  The unit vectors of the coordinate system on the eastward rotating spherical Earth, with $ e_{\varphi} $ pointing in the azimuthal direction
Figure 3.  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
[1]

John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276

[2]

Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416

[3]

Olga A. Brezhneva, Alexey A. Tret’yakov, Jerrold E. Marsden. Higher--order implicit function theorems and degenerate nonlinear boundary-value problems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 293-315. doi: 10.3934/cpaa.2008.7.293

[4]

Hideo Ikeda, Koji Kondo, Hisashi Okamoto, Shoji Yotsutani. On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows. Communications on Pure & Applied Analysis, 2003, 2 (3) : 381-390. doi: 10.3934/cpaa.2003.2.381

[5]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084

[6]

Yang Cao, Wei- Wei Tan, Mei-Qun Jiang. A generalization of the positive-definite and skew-Hermitian splitting iteration. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 811-821. doi: 10.3934/naco.2012.2.811

[7]

Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436

[8]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729

[9]

Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596

[10]

Hongjing Pan, Ruixiang Xing. On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3627-3682. doi: 10.3934/dcds.2015.35.3627

[11]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83

[12]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763

[13]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615

[14]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[15]

Gabriele Bonanno, Pasquale Candito, Roberto Livrea, Nikolaos S. Papageorgiou. Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1169-1188. doi: 10.3934/cpaa.2017057

[16]

John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337

[17]

John R. Graef, Bo Yang. Positive solutions of a third order nonlocal boundary value problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 89-97. doi: 10.3934/dcdss.2008.1.89

[18]

John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269

[19]

Erchuan Zhang, Lyle Noakes. Riemannian cubics and elastica in the manifold $ \operatorname{SPD}(n) $ of all $ n\times n $ symmetric positive-definite matrices. Journal of Geometric Mechanics, 2019, 11 (2) : 277-299. doi: 10.3934/jgm.2019015

[20]

John R. Graef, Lingju Kong. Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian. Conference Publications, 2011, 2011 (Special) : 515-522. doi: 10.3934/proc.2011.2011.515

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (42)
  • HTML views (107)
  • Cited by (0)

Other articles
by authors

[Back to Top]