Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Duchêne, Vincent

doi:10.1051/m2an/2011037

Cited by 19 publications

(42 citation statements)

References 40 publications

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“…Indeed, it suffices to check that a given approximate solution solves our system up to a small remainder, to ensure that it is truly close to the solution of our new model, and therefore to the corresponding solution of the full Euler system. Such strategy has been used in particular in [6] in order to rigorously justify the historical Korteweg-de Vries equation as a model for the propagation of surface wave in the long wave regime (with a Boussinesq model as the intermediary system); and this result has been extended to the bi-fluidic case in [18]. Higher order models in the Camassa-Holm regime have been introduced and justified in the sense of consistency in [12] in the water-wave case, and in [19] in the bi-fluidic case.…”

Section: Presentation Of the Problemmentioning

confidence: 99%

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

Duchêne¹,

Israwi²,

Talhouk³

2015

SIAM J. Math. Anal.

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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.

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Section: Presentation Of the Problemmentioning

confidence: 99%

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

Duchêne¹,

Israwi²,

Talhouk³

2015

SIAM J. Math. Anal.

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“…In that case, one does not expect the free-surface solution to be accurately described by the rigid-lid solution. In other words, the rigid-lid approximation is not valid if is not small; see, for example, the discussion and numerical simulations in [15]. When is small, the essential assumption is the second inequality in (1.5), which can be viewed as an assumption of well-prepared initial data: it ensures that the time-derivative of the flow is initially bounded, uniformly for small.…”

Section: Presentation Of the Models And Main Resultsmentioning

confidence: 99%

“…To our knowledge, very few works are concerned with the validity of the aforementioned approximations, despite the early concerns expressed by Long [31] and Benjamin [4]. Grimshaw, Pelinovsky, Poloukhina [18], Craig, Guyenne, Kalisch [11], Craig, Guyenne, Sulem [12] and the author [15] derived and compared asymptotic models in both the rigid-lid and free-surface settings. However, they do not directly compare solutions of the two models with corresponding initial data, but rather parameters of their models, or explicit solutions (solitary waves).…”

Section: Motivationmentioning

confidence: 99%

On the Rigid-Lid Approximation for Two Shallow Layers of Immiscible Fluids with Small Density Contrast

Duchêne

2014

J Nonlinear Sci

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The rigid-lid approximation is a commonly used simplification in the study of densitystratified fluids in oceanography. Roughly speaking, one assumes that the displacements of the surface are negligible compared with interface displacements. In this paper, we offer a rigorous justification of this approximation in the case of two shallow layers of immiscible fluids with constant and quasi-equal mass density. More precisely, we control the difference between the solutions of the Cauchy problem predicted by the shallow-water (Saint-Venant) system in the rigid-lid and free-surface configuration. We show that in the limit of small density contrast, the flow may be accurately described as the superposition of a baroclinic (or slow) mode, which is well predicted by the rigid-lid approximation; and a barotropic (or fast) mode, whose initial smallness persists for large time. We also describe explicitly the first-order behavior of the deformation of the surface, and discuss the case of non-small initial barotropic mode.

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“…One gets a system of four equations for which the methods of [36] and of the present paper are likely to work, including the case of a slowing varying bottom. We also refer to [17] for further investigations on those extended Boussinesq systems, in particular for a construction of symmetrizable ones (modulo ǫ 2 terms).…”

Section: Long Time Existence For Some Boussinesq Systemsmentioning

confidence: 99%

The Cauchy Problem on Large Time for Surface-Waves-Type Boussinesq Systems II

Saut¹,

Wang²,

Xu³

2017

SIAM J. Math. Anal.

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This paper is a continuation of a previous work by two of the Authors [36] on long time existence for Boussinesq systems modeling the propagation of long, weakly nonlinear water waves. We provide proofs on examples not considered in [36] in particular we prove a long time well-posedness result for a delicate "strongly dispersive" Boussinesq system.where τ ≥ 0 is the surface tension parameter (Bond number).Recall also [6] that (1.2) is linearly well-posed whenand when a = c, b ≥ 0, d ≥ 0. An important step to justify rigorously (1.2) as an asymptotic model for water waves is to establish the well-posedness of the Cauchy problem on time scales of order 1/ǫ, with uniform bounds in suitable Sobolev spaces, the error estimate being then (see [5,29]).This step has been established in [36] (see also [34]) for most of Boussinesq systems with and without surface tension. The idea in [36] is to find an appropriate symmetrization of the system and this is not a straightforward task since one cannot obviously use the classical symmetrizer of the underlying Saint-Venant (shallow water) hyperbolic system. This will be reviewed in the first section of this paper. A complete proof of cases that were not fully developed in [36] will be given in Section 4. 1 Introducing surface tension enlarges the range of physically admissible parameters (a,b,c,d) and so even a local theory 2 for a few linearly well -posed systems is still missing, for instance the cases b = d = 0, a < 0, c = 0 and b = d = 0, a = 0, c < 0). Both cases will be considered here but the later leads to serious difficulties and the long time existence for it is the main result of the present paper.Note also that the (linearly well-posed) "exceptional KdV-KdV" case b = d = 0, a = c > 0 which is studied in [30] leading to well-posedness on time scales of order 1/ √ ǫ in Sobolev spaces H s (R 2 ), s > 3/2 which are larger than the "hyperbolic" one H s (R 2 ), s > 2 is not covered neither in [36] nor in the present paper so that a long time existence is still open in this case 3 .An important mathematical issue concerning Boussinesq systems (1.2) is that despite they describe the same dynamics of water waves, their mathematical properties are rather different, due essentially to their different linear dispersion relations. Of course those dispersion relations all coincide in the long wave limit but there are quite different in the short wave limit. A convenient way to classify the system is according to the order of the Fourier multiplier operator given by the eigenvalues of the linearized operator (see [6]). The order can be −1, 0, 1, 2 or 3. The two last cases are referred to as the strongly dispersive ones.1 Due to the large number of cases to be considered, we chosed in [36] to give complete proofs for a limited number of cases.2 That is not taking care of the dependence of the lifespan of the solution with respect to ǫ 3 However, the case b = d = 0, a < 0, c < 0 that can only occur with a strong surface tension is covered by the theory in [36].

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Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Cited by 19 publications

References 40 publications

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

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

On the Rigid-Lid Approximation for Two Shallow Layers of Immiscible Fluids with Small Density Contrast

The Cauchy Problem on Large Time for Surface-Waves-Type Boussinesq Systems II

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