Asymptotic behaviour of a linearized water waves system in a rectangle¹

Article type: Research Article

Authors: Su, Pei^{; *}

Affiliations: Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351, Cours de la Libération – F 33 405 Talence, France

Correspondence: [*] Corresponding author. E-mail: [email protected].

Note: [1] The author is member of the ETN network ConFlex, funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 765579.

Abstract: We consider the asymptotic behaviour of small-amplitude gravity water waves in a rectangular domain where the water depth is much smaller than the horizontal scale. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a scalar input function u. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ∂ζ∂t. We prove that the solution of the water waves system converges to the solution of the one dimensional wave equation with Neumann boundary control, when taking the shallowness limit. Our approach is based on a special change of variables and a scattering semigroup, which provide the possiblity to apply the Trotter–Kato approximation theorem. Moreover, we use a detailed analysis of Fourier series for the dimensionless version of the partial Dirichlet to Neumann and Neumann to Neumann operators.

Keywords: Linearized water waves equation, Dirichlet to Neumann map, Neumann to Neumann map, operator semigroup, Trotter–Kato theorem

DOI: 10.3233/ASY-221767

Journal: Asymptotic Analysis, vol. 131, no. 1, pp. 83-108, 2023