Well-posedness of the Green–Naghdi and Boussinesq–Peregrine systems

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 [5] 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 Mésognon-Gireau [42]. The main ingredient is a physically motivated change of unknowns revealing the quasilinear structure of the systems, from which energy methods are implemented. * IRMAR -UMR6625, CNRS and Univ. Rennes 1, Campus de Beaulieu, F-35042 Rennes cedex, France. VD is partially supported by the project Dyficolti ANR-13-BS01-0003-01 of the Agence Nationale de la Recherche. † Mathématiques, Faculté des sciences I et Ecole doctorale des sciences et technologie, Université Libanaise, Beyrouth, Liban. SI is partially supported by the Lebanese University research program (MAA group project).

Large-time existence for one-dimensional Green-Naghdi equations with vorticity

“…Remark 3. In [10] the authors show existence and uniqueness of solutions for the two-dimensional Green-Naghdi system in the case of irrotational flows (i.e., with zero vorticity) without the presence of surface tension. The method used therein is different and implies to work with more regular initial data.…”

Modeling shallow water waves

“…without assumption (33)) such as the Boussinesq-Peregrine model (38). Local well posedness for this system has been proved in [70] for times O(1/ max{ε, β}) but the time scale O(1/ε) has only been proved in [144] for a variant of the Boussinesq-Peregrine model (38) tailored to allow the implementation of low Mach techniques developed in [32] for the lake equations.…”

“…This allows one to control the extra nonlinear terms εµQ 1 (h, V ) in (40) which has therefore a semi-linear structure. Local existence was proved in [127] for small times, and in [6,100,81,70] for times of order O(1/ε), uniformly with respect to µ ∈ (0, 1). Another interesting fact shown in [68] is that smooth solutions to the SGN equations can be obtained as relaxation limits of an augmented quasilinear system of conservation laws proposed in [74].…”

“…The Camassa-Holm equation. The equations from the KdV/BBM family (70) are all globally well posed in reasonable Sobolev spaces and therefore unable to reproduce the wave breaking phenomenon. The reason of this is that dispersion balances the nonlinearity.…”

On a shallow-water approximation to the Green–Naghdi equations with the Coriolis effect

“…with an O(µ 2 ) correction, where η(t, x) and u(t, x) are the parameterization of the surface and the vertically averaged horizontal component of the velocity at time t, respectively. A rigorous justification of the GN model can be found in [36] for the 1D water waves with a flat bottom; the general case was handled in [2,20] based on a well-posedness theory.…”