A note on well-posedness of bidirectional Whitham equation

Pei, Long; Wang, Yuexun

doi:10.1016/j.aml.2019.06.015

Cited by 15 publications

(14 citation statements)

References 18 publications

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“…A natural extension of the existing results is to consider the case of non-trivial capillarity κ > 0. Note that the term 1 + κD 2 could be applied to −v x in the first equation instead, as it is done in [12], for example, to regularise the system regarded in [21]. However, the case regarded here is physically more relevant [7].…”

Section: Introductionmentioning

confidence: 99%

Well-Posedness for a Whitham–Boussinesq System with Surface Tension

Dinvay

2020

Math Phys Anal Geom

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We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The proof of well-posedness relies on energy estimates. However, due to the symmetry lack of the nonlinear part, in order to close the a priori estimates one has to modify the traditional energy norm in use. Hamiltonian conservation provides with global well-posedness at least for small initial data in the one dimensional settings.

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Section: Introductionmentioning

confidence: 99%

Well-Posedness for a Whitham–Boussinesq System with Surface Tension

Dinvay

2020

Math Phys Anal Geom

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“…Conservation of Hamiltonian allows to extend globally well-posedness at least for small initial data in the one dimensional case d = 1. This is a nice complement to the existing initial value problem results on other fully dispersive models [31,51].…”

Section: Well-posedness For a Dispersive System Of The Whitham-boussimentioning

confidence: 72%

“…can be found in [15,42]. The corresponding initial value problem was considered in [51] for κ = 0 and in [40] for κ > 0. The latter work provides with a more satisfactory formulation of the Cauchy problem, because of the regularization effect due to the surface tension.…”

Section: The Whitham Equation For Hydroelastic Wavesmentioning

confidence: 99%

“…To our understanding models of such type were introduced mostly ad hoc [1,15,31,39,42]. Moreover, the non-physically conditional well-posedness proved in [51] for one of the systems was not enough. Unfortunately, we were not aware of the system introduced in [31], while writing our paper, because they introduced a fully dispersive system with satisfactory results on the initial value problem [31] and on existence of solitary waves [32].…”

Section: A Comparative Study Of Bi-directional Whitham Systemsmentioning

confidence: 99%

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The Whitham equation for hydroelastic waves

Dinvay

Kalisch

Moldabayev³

et al. 2019

Applied Ocean Research

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The PhD project concerns the surface water wave theory. Liquid is presented as a three or two dimensional layer bounded from below by a rigid horizontal bottom. Above it can have either a free surface or an elastic layer such as an ice cover, for example. The fluid flow is considered to be inviscid, incompressible and irrotational. The flow is described by the Euler equations with some appropriate boundary conditions. Following Lagrange we assume that at the bottom liquid has zero vertical velocity component and at the free surface there is a constant atmospheric pressure. The first condition is very natural meaning that water cannot penetrate through the bottom and is always attached to it making no cavities. The second condition is based on the fact that the air fluctuations are negligible and the pressure is defined by the weight of the atmosphere. Note that it also assumed that the pressure is changing continuously in the air-water media. The latter can be modified in different ways. For example, assuming a strong capillarity effect one can assume that the liquid pressure at the top is proportional to the surface curvature. However, it turns out that this effect is more important in water tanks than in the ocean. Another more relevant modification for the real world is the modelling of ice cover. In such situation the surface tension is caused by deformation forces in the ice. The latter is assumed to be elastic. Solving the Euler equations with the mentioned boundary conditions provides with the complete description of the flow dynamics. However, in many situations it is more important to know only the surface time evolution, whereas solving the full problem may be too demanding. Different nonlinear dispersive wave equations, such as the Korteweg-de Vries equation, allow to approximate the free surface dynamics without providing with the complete description of the fluid motion below the surface. There are several ways of testing the validity of these models. The two most important of them are tests on well-posedness and travelling wave existence. A relevant model should at least reflect adequately these two properties. In this work we are mainly concerned with the fully dispersive bi-directional extensions of the mentioned KdV equation. Still being simple toy models they are believed to be better approximations to the full Euler equations. Moreover, they allow two directional water motion for the two dimensional liquid layer, whereas the scalar KdV equation describes the waves traveling in one direction. A recent interest in such models was caused by discovery of certain phenomena in the Whitham equation that is a direct fully dispersive one-directional extension of the KdV equation. Among them are solitary waves, the existence of a wave of greatest height predicted by Stokes, the existence of shocks and modulational instability of steady periodic waves. A natural step forward is to try to find an adequate two-directional extension of the scalar Whitham equation displaying the same phenomena or some other w...

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“…with m(ξ ) = tanh(ξ )/ξ , which has taken a vast of attractions (we refer to [7] for a fairly complete list of references). There have been several investigations on the system (1.1) with m(ξ ) = tanh(ξ )/ξ [which corresponds to a = −1 in (1.3)]: local well-posedness [13,18], a logarithmically cusped wave of greatest height [6], existence of solitary wave solutions [17], and numerical results [3,4,21]. We should point out that the assumption in [13,18] on the initial surface elevation η 0 ≥ C > 0 is nonphysical, which only yields the well-posedness in homogeneous Sobolev spaces, however the Cauchy problem of (1.1) by this choice of symbol is probably ill-posed for negative initial surface elevation (see a heuristic argument in [13]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Well-Posedness to the Cauchy Problem of a Fully Dispersive Boussinesq System

Wang

2020

J Dyn Diff Equat

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This work concerns the local well-posedness to the Cauchy problem of a fully dispersive Boussinesq system which models fully dispersive water waves in two and three spatial dimensions. Our purpose is to understand the modified energy approach (Kalisch and Pilod in Proc Am Math Soc 147:2545-2559, 2019) in a different point view by utilizing the symmetrization of hyperbolic systems which produces an equivalent modified energy. Keywords Well-posedness • Dispersive Boussinesq system • Symmetrization Mathematics Subject Classification 76B15 • 76B03 • 35S30 in which t ∈ R, x = (x 1 , x 2) ∈ R 2 and the unknowns η ∈ R, v = (v 1 , v 2) ∈ R 2. The operator K is a Fourier multiplier with symbol m ∈ S a ∞ (R d), d = 1 or 2 for some a ∈ R\{0}, Y.W. acknowledges the support by Grants Nos. 231668 and 250070 from the Research Council of Norway.

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A note on well-posedness of bidirectional Whitham equation

Cited by 15 publications

References 18 publications

Well-Posedness for a Whitham–Boussinesq System with Surface Tension

Well-Posedness for a Whitham–Boussinesq System with Surface Tension

The Whitham equation for hydroelastic waves

Well-Posedness to the Cauchy Problem of a Fully Dispersive Boussinesq System

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