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Navier's slip and incompressible limits in domains with variable bottoms
A mathematical review of the analysis of the betaplane model and equatorial waves
| 1. | Institut de Mathématiques de Jussieu UMR 7586, Université Paris VII, 175, rue du Chevaleret, 75013 Paris, France |
More precisely we are interested in the study of the shallow water equations set in the vicinity of the equator: in that situation the Coriolis force vanishes and its linearization near zero leads to the so-called betaplane model. Our aim is to study the asymptotics of this model in the limit of small Rossby and Froude numbers. We show in a first part the existence and uniqueness of bounded (strong) solutions on a uniform time, and we study their weak limit. In a second part we give a more precise account of the asymptotics by characterizing the possible defects of compactness to that limit, in the framework of weak solutions only.
These results are based on the studies [6]-[8] on the one hand, and [11] on the other.
| [1] |
Elena Kartashova. Nonlinear resonances of water waves. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607 |
| [2] |
Anatoly Abrashkin. Wind generated equatorial Gerstner-type waves. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4443-4453. doi: 10.3934/dcds.2019181 |
| [3] |
Tony Lyons. Geophysical internal equatorial waves of extreme form. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4471-4486. doi: 10.3934/dcds.2019183 |
| [4] |
Jifeng Chu, Joachim Escher. Steady periodic equatorial water waves with vorticity. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4713-4729. doi: 10.3934/dcds.2019191 |
| [5] |
David Henry, Hung-Chu Hsu. Instability of equatorial water waves in the $f-$plane. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 909-916. doi: 10.3934/dcds.2015.35.909 |
| [6] |
David Henry. Energy considerations for nonlinear equatorial water waves. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2337-2356. doi: 10.3934/cpaa.2022057 |
| [7] |
Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41 |
| [8] |
Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75 |
| [9] |
Shu-Lin Lyu. On the Hermite--Hadamard inequality for convex functions of two variables. Numerical Algebra, Control and Optimization, 2014, 4 (1) : 1-8. doi: 10.3934/naco.2014.4.1 |
| [10] |
Tomasz Szostok. Inequalities of Hermite-Hadamard type for higher order convex functions, revisited. Communications on Pure and Applied Analysis, 2021, 20 (2) : 903-914. doi: 10.3934/cpaa.2020296 |
| [11] |
Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165 |
| [12] |
Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045 |
| [13] |
S. S. Dragomir, I. Gomm. Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 271-278. doi: 10.3934/naco.2012.2.271 |
| [14] |
Biswajit Basu. On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4783-4796. doi: 10.3934/dcds.2019195 |
| [15] |
Tony Lyons. Particle paths in equatorial flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2399-2414. doi: 10.3934/cpaa.2022041 |
| [16] |
Ruonan Liu, Run Xu. Hermite-Hadamard type inequalities for harmonical $ (h1,h2)- $convex interval-valued functions. Mathematical Foundations of Computing, 2021, 4 (2) : 89-103. doi: 10.3934/mfc.2021005 |
| [17] |
Xiao-Qiang Zhao, Wendi Wang. Fisher waves in an epidemic model. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1117-1128. doi: 10.3934/dcdsb.2004.4.1117 |
| [18] |
Darryl D. Holm, Cornelia Vizman. Dual pairs in resonances. Journal of Geometric Mechanics, 2012, 4 (3) : 297-311. doi: 10.3934/jgm.2012.4.297 |
| [19] |
Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems and Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225 |
| [20] |
Kazuyuki Yagasaki. Degenerate resonances in forced oscillators. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 423-438. doi: 10.3934/dcdsb.2003.3.423 |
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