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On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density
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On stratified water waves with critical layers and Coriolis forces
Study of a nonlinear boundary-value problem of geophysical relevance
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria |
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.
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.![]() ![]() |



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