On a nonlinear integral equation for the ocean flow in arctic gyres
Author:
Jifeng Chu
Journal:
Quart. Appl. Math. 76 (2018), 489-498
MSC (2010):
Primary 45G99, 58J32, 76B03
DOI:
https://doi.org/10.1090/qam/1486
Published electronically:
September 19, 2017
MathSciNet review:
3805038
Full-text PDF
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Additional Information
Abstract: We investigate the existence and uniqueness of bounded solutions to a nonlinear integral equation which models the ocean flow in arctic gyres.
- R. P. Agarwal and V. Lakshmikantham, Uniqueness and nonuniqueness criteria for ordinary differential equations, Series in Real Analysis, vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1336820
- Jifeng Chu and Pedro J. Torres, Applications of Schauder’s fixed point theorem to singular differential equations, Bull. Lond. Math. Soc. 39 (2007), no. 4, 653–660. MR 2346946, DOI https://doi.org/10.1112/blms/bdm040
- Jifeng Chu, Pedro J. Torres, and Meirong Zhang, Periodic solutions of second order non-autonomous singular dynamical systems, J. Differential Equations 239 (2007), no. 1, 196–212. MR 2341553, DOI https://doi.org/10.1016/j.jde.2007.05.007
- J. Chu, On a differential equation arising in geophysics, Monatsh. Math. https://doi.org/10.1007/ s00605-017-1087-1.
- J. Chu, On a nonlinear model for arctic gyres, Ann. Mat. Pura Appl. (4), https://doi.org/10.1007/ s10231-017-0696-6.
- Adrian Constantin, On the existence of positive solutions of second order differential equations, Ann. Mat. Pura Appl. (4) 184 (2005), no. 2, 131–138. MR 2149089, DOI https://doi.org/10.1007/s10231-004-0100-1
- Adrian Constantin, A note on the uniqueness of solutions of ordinary differential equations, Appl. Anal. 64 (1997), no. 3-4, 273–276. MR 1460083, DOI https://doi.org/10.1080/00036819708840535
- A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn. 109 (2015), no. 4, 311–358. MR 3375654, DOI https://doi.org/10.1080/03091929.2015.1066785
- A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr. 46 (2016), 1935–1945.
- 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.
- A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates, Proc. A. 473 (2017), no. 2200, 20170063, 17. MR 3650591, DOI https://doi.org/10.1098/rspa.2017.0063
- A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Physics of Fluids, 29 (2017), 056604.
- A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech. 820 (2017), 511–528. MR 3659720, DOI https://doi.org/10.1017/jfm.2017.223
- W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965. MR 0190463
- C. Corduneanu, Integral equations and applications, Cambridge University Press, Cambridge, 1991. MR 1109491
- R. S. Johnson, An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics, J. Nonlinear Math. Phys. 22 (2015), no. 4, 475–493. MR 3434074, DOI https://doi.org/10.1080/14029251.2015.1113042
- Eberhard Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR 816732
- R. P. Agarwal and V. Lakshmikantham, Uniqueness and nonuniqueness criteria for ordinary differential equations, Series in Real Analysis, vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1336820
- J. Chu and P. J. Torres, Applications of Schauder’s fixed point theorem to singular differential equations, Bull. Lond. Math. Soc. 39 (2007), no. 4, 653–660. MR 2346946, DOI https://doi.org/10.1112/blms/bdm040
- J. Chu, P. J. Torres, and M. Zhang, Periodic solutions of second order non-autonomous singular dynamical systems, J. Differential Equations 239 (2007), no. 1, 196–212. MR 2341553, DOI https://doi.org/10.1016/j.jde.2007.05.007
- J. Chu, On a differential equation arising in geophysics, Monatsh. Math. https://doi.org/10.1007/ s00605-017-1087-1.
- J. Chu, On a nonlinear model for arctic gyres, Ann. Mat. Pura Appl. (4), https://doi.org/10.1007/ s10231-017-0696-6.
- A. Constantin, On the existence of positive solutions of second order differential equations, Ann. Mat. Pura Appl. (4) 184 (2005), no. 2, 131–138. MR 2149089, DOI https://doi.org/10.1007/s10231-004-0100-1
- A. Constantin, A note on the uniqueness of solutions of ordinary differential equations, Appl. Anal. 64 (1997), no. 3-4, 273–276. MR 1460083, DOI https://doi.org/10.1080/00036819708840535
- A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn. 109 (2015), no. 4, 311–358. MR 3375654, DOI https://doi.org/10.1080/03091929.2015.1066785
- A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr. 46 (2016), 1935–1945.
- 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.
- A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates, Proc. A. 473 (2017), no. 2200, 20170063, 17. MR 3650591
- A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Physics of Fluids, 29 (2017), 056604.
- A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech. 820 (2017), 511–528. MR 3659720, DOI https://doi.org/10.1017/jfm.2017.223
- W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965. MR 0190463
- C. Corduneanu, Integral equations and applications, Cambridge University Press, Cambridge, 1991. MR 1109491
- R. S. Johnson, An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics, J. Nonlinear Math. Phys. 22 (2015), no. 4, 475–493. MR 3434074, DOI https://doi.org/10.1080/14029251.2015.1113042
- E. Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR 816732
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Additional Information
Jifeng Chu
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
MR Author ID:
722681
Email:
jifengchu@126.com, jchu@shnu.edu.cn
Received by editor(s):
July 26, 2017
Received by editor(s) in revised form:
August 16, 2017
Published electronically:
September 19, 2017
Additional Notes:
This work was supported by the National Natural Science Foundation of China (Grant No. 11671118)
Article copyright:
© Copyright 2017
Brown University