Asymptotic behaviour of a linearized water waves system in a rectangle
| Autor: | SU, Pei |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
[PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn]
Fluid Dynamics (physics.flu-dyn) FOS: Physical sciences Physics - Fluid Dynamics [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] Dirichlet to Neumann map Operator semigroup Functional Analysis (math.FA) Mathematics - Functional Analysis Neumann to Neumann map Mathematics - Analysis of PDEs Optimization and Control (math.OC) FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Trotter-Kato theorem [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] Mathematics - Optimization and Control Linearized water waves equation Analysis of PDEs (math.AP) |
| Zdroj: | HAL |
| Popis: | 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 $\zeta$ along the top boundary, and its time derivative $\partial$$\zeta$ $\partial$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 introduced in [1]. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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