On the two-dimensional tidal dynamics system: stationary solution and stability. (English) Zbl 1456.76138

Summary: In this work, we consider the two-dimensional stationary and non-stationary tidal dynamic equations and examine the asymptotic behavior of the stationary solution. We prove the existence and uniqueness of weak and strong solutions of the stationary tidal dynamic equations in bounded domains using compactness arguments. Using maximal monotonicity property of the linear and nonlinear operators, we also establish that the solvability results are even valid in unbounded domains. Later, we obtain a uniform Lyapunov stability of the steady state solution. Finally, we remark that the stationary solution is exponentially stable if we add a suitable dissipative term in the equation corresponding to the deviations of free surface with respect to the ocean bottom. This exponential stability helps us to ensure the mass conservation of the modified system, if we choose the initial data of the modified system as stationary solution.


76U60 Geophysical flows
76E20 Stability and instability of geophysical and astrophysical flows
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI


[1] Galilei, G., Dialogue concerning the two chief world systems (1967), Berkeley, CA: University of California Press, Berkeley, CA
[2] Newton, I., Philosophiae naturalis principia mathematica (1687), London: Encyclopædia Britannica, London
[3] Marchuk, GI; Kagan, BA., Ocean tides: mathematical models and numerical experiments (1984), Elmsford (NY): Pergamon Press, Elmsford (NY)
[4] Marchuk, GI; Kagan, BA., Dynamics of ocean tides (1989), Dordrecht/Boston/London: Kluwer Academic Publishers, Dordrecht/Boston/London
[5] Kagan, BA., Hydrodynamic models of tidal motions in the sea (Russian) (1968), Leningrad: Gidrometeoizdat, Leningrad
[6] Gordeev, RG., The existence of a periodic solution in tide dynamic problem, J Soviet Math, 6, 1, 1-4 (1976)
[7] Manna, U.; Menaldi, JL; Sritharan, SS., Stochastic analysis of tidal dynamics equation, Infin Dimens Stoch Anal, 90-113 (2008) · Zbl 1144.86302
[8] Yin, H., Stochastic analysis of backward tidal dynamics equation, Commun Stochastic Anal, 5, 745-768 (2011) · Zbl 1331.60124
[9] Ipatova, VM., Solvability of a tide dynamics model in adjacent seas, Russian J Numer Anal Math Modelling, 20, 1, 67-79 (2005) · Zbl 1084.86004
[10] Suvinthra, M.; Sritharan, SS; Balachandran, K., Large deviations for stochastic tidal dynamics equations, Commun Stoch Anal, 9, 4, 477-502 (2015)
[11] Agarwal, P.; Manna, U.; Mukherjee, D., Stochastic control of tidal dynamics equation with Lévy noise, Appl Math Optim, 1-70 (2017) · Zbl 1420.35220
[12] Haseena, A, Suvinthra, M, Mohan, MT, et al. Moderate deviations for stochastic tidal dynamics equation with multiplicative noise. Submitted.
[13] Pedlosky, J., Geophysical fluid dyanmics I, II (1981), Heidelberg: Springer, Heidelberg
[14] Lyapunov, AM.The general problem of the stability of motion (In Russian) [doctoral dissertation]. Univ. Kharkov; 1892. · Zbl 0041.32204
[15] Barbu, V., Stabilization of Navier-Stokes flows (2011), London: Springer-Verlag · Zbl 1213.76001
[16] Temam, R., Navier-stokes equations (1984), North-Holland, Amsterdam: Theory and Numerical Analysis, North-Holland, Amsterdam
[17] DiBenedetto, E., Degenerate parabolic equations (1993), New York: Springer-Verlag, New York · Zbl 0794.35090
[18] Ladyzhenskaya, OA., The mathematical theory of viscous incompressible flow (1969), New York: Gordon and Breach, New York
[19] Evans, LC.Partial differential equations. Grad. Stud. Math. vol. 19. Amer. Math. Soc. Providence (RI); 1998.
[20] Galdi, GP., An introduction to the mathematical theory of the Navier-Stokes equations (1994), New York: Springer-Verlag, Inc., New York
[21] Temam, R., Navier-Stokes equations and nonlinear functional analysis (1983), Philadelphia: SIAM, Philadelphia · Zbl 0522.35002
[22] Smith, W.; Stegenga, DA., Hölder domains and Poincaré domains, Trans Amer Math Soc, 319, 1, 67-100 (1990) · Zbl 0707.46028
[23] Barbu, V., Analysis and control of nonlinear infinite dimensional systems, Vol. 190 (1993), Boston (MA): Academic Press Inc., Boston (MA)
[24] Cioranescu, I., Geometry of banach spaces, duality mapping and nonlinear problems (1990), Boston: Kluwer Academic, Boston
[25] Roubiuaek, T., Nonlinear partial differential equations with applications (2013), Basel: Birkhüser, Basel
[26] Zeidler, E., Nonlinear functional analysis and its applications II/B (1990), New York: Springer, New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.