In this paper, we derive asymptotic models for the propagation of two and threedimensional gravity waves at the free surface and the interface between two layers of immiscible fluids of different densities, over an uneven bottom. We assume the thickness of the upper and lower fluids to be of comparable size, and small compared to the characteristic wavelength of the system (shallow water regimes). Following a method introduced by Bona, Lannes and Saut based on the expansion of the involved Dirichlet-to-Neumann operators, we are able to give a rigorous justification of classical models for weakly and strongly nonlinear waves, as well as interesting new ones. In particular, we derive linearly well-posed systems in the so called Boussinesq/Boussinesq regime. Furthermore, we establish the consistency of the full Euler system with these models, and deduce the convergence of the solutions.

We introduce a new class of Green-Naghdi type models for the propagation of internal waves between two (1 + 1)-dimensional layers of homogeneous, immiscible, ideal, incompressible, irrotational fluids, vertically delimited by a flat bottom and a rigid lid. These models are tailored to improve the frequency dispersion of the original bi-layer Green-Naghdi model, and in particular to manage high-frequency Kelvin-Helmholtz instabilities, while maintaining its precision in the sense of consistency. Our models preserve the Hamiltonian structure, symmetry groups and conserved quantities of the original model. We provide a rigorous justification of a class of our models thanks to consistency, well-posedness and stability results. These results apply in particular to the original Green-Naghdi model as well as to the Saint-Venant (hydrostatic shallow-water) system with surface tension.

This study deals with asymptotic models for the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. We present a new Green-Naghdi type model in the Camassa-Holm (or medium amplitude) regime. This model is fully justified, in the sense that it is consistent, well-posed, and that its solutions remain close to exact solutions of the full Euler system with corresponding initial data. Moreover, our system allows to fully justify any well-posed and consistent lower order model; and in particular the so-called Constantin-Lannes approximation, which extends the classical Korteweg-de Vries equation in the Camassa-Holm regime. (convergence) is the difference between these two solutions small over the relevant time scale?As mentioned earlier, Lannes has recently proved [29] that the Cauchy problem for bi-fluidic full Euler system is well-posed in Sobolev spaces, in the presence of a small amount of surface tension. Thus the full justification of a consistent system of equation as an asymptotic model, in the sense described above, follows from its well-posedness and a stability result; see [27, Appendix C] for a detailed discussion and state of the art in the water-wave setting.A striking discrepancy between the water-wave and the bi-fluidic setting is that in the latter, large amplitude internal waves are known to generate Kelvin-Helmholtz instabilities, so that surface tension is necessary in order to regularize the flow. A crucial contribution of [29] consists in asserting that "the Kelvin-Helmholtz instabilities appear above a frequency threshold for which surface tension is relevant, while the main (observable) part of the wave involves low frequencies located below this frequency threshold". It is therefore expected that the surface tension does not play an essential role in the dominant evolution of the flow, especially in the shallow water regime. This intuition is confirmed by the fact that well-posedness and stability results have been proved for the bi-fluidic shallow-water system [22], and a class of Boussinesq-type systems [18], without surface tension and under reasonable assumptions on the flow (typically, the shear velocity must be sufficiently small). However, the original bi-fluidic Green-Naghdi model is known to be unconditionally ill-posed [30], which has led to various propositions in order to overcome this difficulty; see [3,13] and references therein. Let us recall here that Green-Naghdi models consist in higher order extensions of the shallow water equation, thus are consistent with precision O(µ 2 ) instead of O(µ), and allow strong nonlinearities (whereas Boussinesq models are limited to the long wave regime: = O(µ)). Finally, we mention the work of Xu [39], which studies and rigorously justifies the so-called intermediate long wave system, obtained in a regime similar to ours: ∼ √ µ, but δ ∼ √ µ.

Abstract. We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Choi and R. Camassa, J. Fluid Mech. 313 (1996) 83-103]. We study the wellposedness of such systems, and the asymptotic convergence of their solutions towards solutions of the full Euler system. Then, we provide a rigorous justification of the so-called KdV approximation, stating that any bounded solution of the full Euler system can be decomposed into four propagating waves, each of them being well approximated by the solutions of uncoupled Korteweg-de Vries equations. Our method also applies for models with the rigid lid assumption, using the Boussinesq/Boussinesq models introduced in [J.L. Bona, D. Lannes and J.-C. Saut, J. Math. Pures Appl. 89 (2008) 538-566]. Our explicit and simultaneous decomposition allows to study in details the behavior of the flow depending on the depth and density ratios, for both the rigid lid and free surface configurations. In particular, we consider the influence of the rigid lid assumption on the evolution of the interface, and specify its domain of validity. Finally, solutions of the Boussinesq/Boussinesq systems and the KdV approximation are numerically computed, using a Crank-Nicholson scheme with a predictive step inspired from [C.

We investigate scattering, localization, and dispersive time decay properties for the one‐dimensional Schrödinger equation with a rapidly oscillating and spatially localized potential q ε = q ( x , x / ε ), where q ( x , y ) is periodic and mean zero with respect to y. Such potentials model a microstructured medium. Homogenization theory fails to capture the correct low‐energy (k small) behavior of scattering quantities, e.g., the transmission coefficient t q ε ( k ) as ∊ tends to zero. We derive an effective potential well σeffε ( x ) = − ε 2 Λ eff ( x ) such that t q ε ( k ) − t σeffε ( k ) is small, uniformly for k ∈ R as well as in any bounded subset of a suitable complex strip. Within such a bounded subset, the scaled limit of the transmission coefficient has a universal form, depending on a single parameter, which is computable from the effective potential. A consequence is that if ϵ, the scale of oscillation of the microstructure potential, is sufficiently small, then there is a pole of the transmission coefficient (and hence of the resolvent) in the upper half‐plane on the imaginary axis at a distance of order ε 2 from 0. It follows that the Schrödinger operator H q ε = − ∂ x 2 + q ε ( x ) has an L 2 bound state with negative energy situated a distance frakturO ( ε 4 ) from the edge of the continuous spectrum. Finally, we use this detailed information to prove the local energy time decay estimate: | ( 1 + | · | ) − 3 e − i t H q e P c ψ 0 | L ∞ ≤ * C t − 1 / 2 true( 1 + e 4   true( ∫ l ∫ R Λ eff ) 2 t ) − 1 | ( 1 + | · | 3 ) ψ 0 | L 1 , where P c denotes the projection onto the continuous spectral part of H q ε. © 2013 Wiley Periodicals, Inc.