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Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity
Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows
1. | Department of Mathematics, Guizhou University, Guiyang 550025, China |
2. | College of Mathematics and Information Science, Guiyang University, Guiyang 550005, China |
3. | Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia |
4. | Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49,814 73 Bratislava, Slovakia |
5. | School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China |
In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with constant eddy viscosity. Different from the well-known homogeneous system in [
References:
[1] |
C. Alsina and R. Ger,
On some inequalities and stability results related to the exponential function, J. Inequal. Appl., 2 (1998), 373-380.
doi: 10.1155/S102558349800023X. |
[2] |
S. Blanes and F. Casas,
On the convergence and optimization of the Baker-Campbell-Hausdorff formula, Linear Algebra Appl., 378 (2004), 135-158.
doi: 10.1016/j.laa.2003.09.010. |
[3] |
A. Bressan and A. Constantin,
The deflection angle of surface ocean currents from the wind direction, J. Geophys. Res. Oceans, 124 (2019), 7412-7420.
doi: 10.1029/2019JC015454. |
[4] |
C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 1999. |
[5] |
A. Constantin and R. I. Ivanov,
Equatorial wave-current interactions, Comm. Math. Phys, 370 (2019), 1-48.
doi: 10.1007/s00220-019-03483-8. |
[6] |
A. Constantin and R. S. Johnson,
Atmospheric Ekman flows with variable eddy viscosity, Boundary-Layer Meteorology, 170 (2019), 395-414.
doi: 10.1007/s10546-018-0404-0. |
[7] |
A. Constantin and R. S. Johnson,
On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr, 49 (2019), 2029-2042.
doi: 10.1175/JPO-D-19-0079.1. |
[8] |
W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, 629, Springer-Verlag, Berlin/New York, 1978.
doi: 10.1007/BFb0067780. |
[9] |
V. W. Ekman, On the influence of the Earth's rotation on ocean-currents, Arkiv Matematik Astronmi Och Fysik, 2 (1905), 1-52. Google Scholar |
[10] |
M. Fečkan, Y. Guan, D. O'Regan and J. Wang, Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows, Monatshefte für Mathematik, 115 (2020).
doi: 10.1007/s00605-020-01414-7. |
[11] |
B. Grisogono,
A generalized Ekman layer profile with gradually varying eddy diffusivities, Quart. J. Roy. Meteor. Soc., 121 (1995), 445-453.
doi: 10.1002/qj.49712152211. |
[12] | G. J. Haltinar and R. T. Williams, Numerical Prediction and Dynamic Meteorology, Wiley Press, New York, 1980. Google Scholar |
[13] |
D. Henry,
Nonlinear features of equatorial ocean flows, Oceanography, 31 (2018), 22-27.
doi: 10.5670/oceanog.2018.305. |
[14] | J. R. Holton, An Introduction to Dynamic Meteorology, Academic Press, New York, 2004. Google Scholar |
[15] |
D. H. Hyers,
On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224.
doi: 10.1073/pnas.27.4.222. |
[16] |
S. M. Jung,
Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl., 320 (2006), 549-561.
doi: 10.1016/j.jmaa.2005.07.032. |
[17] |
S. M. Jung,
Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17 (2004), 1135-1140.
doi: 10.1016/j.aml.2003.11.004. |
[18] |
S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Inc., Palm Harbor, FL, 2001. |
[19] |
O. S. Madsen,
A realistic model of the wind-induced Ekman boundary layer, J. Phys. Oceanogr., 7 (1977), 248-255.
doi: 10.1175/1520-0485(1977)007<0248:ARMOTW>2.0.CO;2. |
[20] | J. Marshall and R. A. Plumb, Atmosphere, Ocean and Climate Dynamic, Academic Press, New York, 2008. Google Scholar |
[21] |
F. T. M. Nieuwstadt,
On the solution of the stationary, baroclinic Ekman-layer equations with a finite boundary-layer height, Boundary-Layer Meteorology, 26 (1983), 377-390.
doi: 10.1007/BF00119534. |
[22] |
O. Parmhed, I. Kos and B. Grisogono,
An improved Ekman layer approximation for smooth eddy diffusivity profiles, Boundary-Layer Meteorology, 115 (2005), 399-407.
doi: 10.1007/s10546-004-5940-0. |
[23] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag Press, New York, 1987.
doi: 10.1007/978-1-4612-4650-3.![]() |
[24] |
J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, 59, Springer, New York, 2007.
doi: 10.1007/978-0-387-48918-6. |
[25] |
A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1980. |
[26] |
J. Wang and M. Fečkan,
Ulam-Hyers-Rassias stability for semilinear equations, Discontinuity Nonlinear Complexity, 3 (2014), 379-388.
doi: 10.5890/DNC.2014.12.002. |
[27] |
J. O. Wenegrat and M. J. McPhaden,
Wind, waves, and fronts: Frictional effects in a generalized Ekman model, J. Phys. Oceanogr., 46 (2016), 371-394.
doi: 10.1175/JPO-D-15-0162.1. |
[28] |
W. Zdunkowski and A. Bott, Dynamic of the Atmosphere, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511805462.![]() |
show all references
References:
[1] |
C. Alsina and R. Ger,
On some inequalities and stability results related to the exponential function, J. Inequal. Appl., 2 (1998), 373-380.
doi: 10.1155/S102558349800023X. |
[2] |
S. Blanes and F. Casas,
On the convergence and optimization of the Baker-Campbell-Hausdorff formula, Linear Algebra Appl., 378 (2004), 135-158.
doi: 10.1016/j.laa.2003.09.010. |
[3] |
A. Bressan and A. Constantin,
The deflection angle of surface ocean currents from the wind direction, J. Geophys. Res. Oceans, 124 (2019), 7412-7420.
doi: 10.1029/2019JC015454. |
[4] |
C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 1999. |
[5] |
A. Constantin and R. I. Ivanov,
Equatorial wave-current interactions, Comm. Math. Phys, 370 (2019), 1-48.
doi: 10.1007/s00220-019-03483-8. |
[6] |
A. Constantin and R. S. Johnson,
Atmospheric Ekman flows with variable eddy viscosity, Boundary-Layer Meteorology, 170 (2019), 395-414.
doi: 10.1007/s10546-018-0404-0. |
[7] |
A. Constantin and R. S. Johnson,
On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr, 49 (2019), 2029-2042.
doi: 10.1175/JPO-D-19-0079.1. |
[8] |
W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, 629, Springer-Verlag, Berlin/New York, 1978.
doi: 10.1007/BFb0067780. |
[9] |
V. W. Ekman, On the influence of the Earth's rotation on ocean-currents, Arkiv Matematik Astronmi Och Fysik, 2 (1905), 1-52. Google Scholar |
[10] |
M. Fečkan, Y. Guan, D. O'Regan and J. Wang, Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows, Monatshefte für Mathematik, 115 (2020).
doi: 10.1007/s00605-020-01414-7. |
[11] |
B. Grisogono,
A generalized Ekman layer profile with gradually varying eddy diffusivities, Quart. J. Roy. Meteor. Soc., 121 (1995), 445-453.
doi: 10.1002/qj.49712152211. |
[12] | G. J. Haltinar and R. T. Williams, Numerical Prediction and Dynamic Meteorology, Wiley Press, New York, 1980. Google Scholar |
[13] |
D. Henry,
Nonlinear features of equatorial ocean flows, Oceanography, 31 (2018), 22-27.
doi: 10.5670/oceanog.2018.305. |
[14] | J. R. Holton, An Introduction to Dynamic Meteorology, Academic Press, New York, 2004. Google Scholar |
[15] |
D. H. Hyers,
On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224.
doi: 10.1073/pnas.27.4.222. |
[16] |
S. M. Jung,
Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl., 320 (2006), 549-561.
doi: 10.1016/j.jmaa.2005.07.032. |
[17] |
S. M. Jung,
Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17 (2004), 1135-1140.
doi: 10.1016/j.aml.2003.11.004. |
[18] |
S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Inc., Palm Harbor, FL, 2001. |
[19] |
O. S. Madsen,
A realistic model of the wind-induced Ekman boundary layer, J. Phys. Oceanogr., 7 (1977), 248-255.
doi: 10.1175/1520-0485(1977)007<0248:ARMOTW>2.0.CO;2. |
[20] | J. Marshall and R. A. Plumb, Atmosphere, Ocean and Climate Dynamic, Academic Press, New York, 2008. Google Scholar |
[21] |
F. T. M. Nieuwstadt,
On the solution of the stationary, baroclinic Ekman-layer equations with a finite boundary-layer height, Boundary-Layer Meteorology, 26 (1983), 377-390.
doi: 10.1007/BF00119534. |
[22] |
O. Parmhed, I. Kos and B. Grisogono,
An improved Ekman layer approximation for smooth eddy diffusivity profiles, Boundary-Layer Meteorology, 115 (2005), 399-407.
doi: 10.1007/s10546-004-5940-0. |
[23] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag Press, New York, 1987.
doi: 10.1007/978-1-4612-4650-3.![]() |
[24] |
J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, 59, Springer, New York, 2007.
doi: 10.1007/978-0-387-48918-6. |
[25] |
A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1980. |
[26] |
J. Wang and M. Fečkan,
Ulam-Hyers-Rassias stability for semilinear equations, Discontinuity Nonlinear Complexity, 3 (2014), 379-388.
doi: 10.5890/DNC.2014.12.002. |
[27] |
J. O. Wenegrat and M. J. McPhaden,
Wind, waves, and fronts: Frictional effects in a generalized Ekman model, J. Phys. Oceanogr., 46 (2016), 371-394.
doi: 10.1175/JPO-D-15-0162.1. |
[28] |
W. Zdunkowski and A. Bott, Dynamic of the Atmosphere, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511805462.![]() |
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