On the two-dimensional tidal dynamics system: stationary solution and stability

ABSTRACT

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.

Keywords:

• Tidal dynamics system
• stationary tidal dynamic equations
• weak and strong solutions
• monotonicity
• stability

Mathematics Subject Classification (2010):

• 76D03
• 74G25
• 37L15

Acknowledgements

The author sincerely would like to thank the reviewers for their valuable comments and suggestions, which led to the improvement of this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The author would like to thank the Department of Science and Technology (DST), Govt of India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110) and Indian Institute of Technology Roorkee-IIT Roorkee, for providing stimulating scientific environment and resources.