Abstract
In this paper we address the Cauchy problem for two systems modeling the propagation of long gravity waves in a layer of homogeneous, incompressible and inviscid fluid delimited above by a free surface, and below by a non-necessarily flat rigid bottom. Concerning the Green-Naghdi system, we improve the result of Alvarez-Samaniego and Lannes (Invent. Math., 2008) in the sense that much less regular data are allowed, and no loss of derivatives is involved. Concerning the Boussinesq-Peregrine system, we improve the lower bound on the time of existence provided by M{\'e}sognon-Gireau (Adv. Differential Equations, 2017). The main ingredient is a physically motivated change of unknowns revealing the quasilinear structure of the systems, from which energy methods are implemented.
Highlights
For a clear and modern exposition, it is shown in [32] that the Green–Naghdi system can be derived as an asymptotic model from the water waves system, by assuming that the typical horizontal length of the flow is much larger than the depth of the fluid layer and that the flow is irrotational
The derivation through formal Taylor expansions of Bonneton and Lannes [32] can be made rigorous [31, Prop. 5.8]: roughly speaking, any sufficiently smooth solution of the water waves system satisfies the Green–Naghdi system up to a quantifiable remainder. This consistency result is only one step towards the full justification of the model in the following sense: the solution of the water waves system and the solution of the Green–Naghdi system with corresponding initial data remain close on a relevant time interval
The main result of this paper is to show that this loss of derivatives is not necessary, WP of Green–Naghdi and Boussinesq–Peregrine and that the Cauchy problem for the Green–Naghdi system is well-posed in the sense of Hadamard in Sobolev-type spaces
Summary
The Green–Naghdi system (sometimes called Serre or fully nonlinear Boussinesq system) is a model for the propagation of surface gravity waves in a layer of homogeneous incompressible inviscid fluid with rigid bottom and free surface It has been formally derived several times in the literature, in particular in [25, 42, 46, 47, 50], using different techniques and various hypotheses. 5.8]: roughly speaking, any sufficiently smooth solution of the water waves system satisfies the Green–Naghdi system up to a quantifiable (small) remainder This consistency result is only one step towards the full justification of the model in the following sense: the solution of the water waves system and the solution of the Green–Naghdi system with corresponding initial data remain close on a relevant time interval. The main result of this paper is to show that this loss of derivatives is not necessary, WP of Green–Naghdi and Boussinesq–Peregrine and that the Cauchy problem for the Green–Naghdi system is well-posed in the sense of Hadamard in Sobolev-type spaces
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
