A New Fully Justified Asymptotic Model for the Propagation of Internal Waves in the Camassa--Holm Regime

Abstract

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.

A New Class of Two‐Layer Green–Naghdi Systems with Improved Frequency Dispersion

“…Many attempts have been made to "regularize" the Green-Naghdi model, that is proposing new models with formally the same precision as the original model, but which are not subject to high-frequency Kelvin-Helmholtz instabilities, even without surface tension [11][12][13][14][15]. The strategies adopted in these works rely on change of unknowns and/or Benjamin-Bona-Mahony type tricks; see [16, section 5.2] for a thorough presentation of such methods in the free-surface setting.…”

Comparison of nonlocal nonlinear wave equations in the long-wave limit

“…We note that, in the terminology of some authors, our long-time existence results [2] together with the work presented here are in fact consistency, existence, convergence results for the nonlocal bidirectional approximations of the nonlocal equation (1). We refer to [3][4][5][6][7] and the references therein for a detailed discussion of these concepts.…”

“…for some c 1 , c 2 > 0. The condition (7) implies that the convolution operator is invertible. On the other hand, (7) lacks the regularization effect of (6) for r ≥ 2.…”

“…The computation in [2] shows that the optimal values of D and P are approximately 7.35 and 55.34, respectively. Throughout this work we will assume that the kernel β(x) is an even function with β(x)dx = 1 and satisfies (7).…”

“…where k(ξ) = β(ξ) and F −1 denotes the inverse Fourier transform. We note that K 2 w = β * w and, by (7), K is an invertible bounded operator on H s . We then convert (5) to…”