Rigorous Derivation from the Water Waves Equations of Some Full Dispersion Shallow Water Models

Emerald, Louis

doi:10.1137/20m1332049

Cited by 15 publications

(16 citation statements)

References 17 publications

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“…Suppose to the contrary that there exists a sequence {(ϕ j , c j )} j of such solution pairs for which c j ց 0. Then n ′ (ϕ j (−π)) c j ց 0 as well, contradicting (13). Thus c 1 uniformly and n ′ (ϕ(−π)) does not touch c, so ϕ is smooth around −π by Lemma 3.3.…”

Section: /31mentioning

confidence: 87%

“…Full-dispersion nonlinear evolution equations such as (1) have seen a keen interest in the recent years as nonlocal improvements of classical local equations. In particular, surfacewave models in shallow water of this class with various dispersive operators approximate the full water-wave equations [13,14] and capture singular features not found in their local counterparts.…”

Section: Introductionmentioning

confidence: 99%

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Periodic Hölder waves in a class of negative-order dispersive equations

Hildrum¹,

Xue²

2022

Preprint

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A. We prove the existence of highest, cusped, periodic travelling-wave solutions with exact and optimal α-Hölder continuity in a family of negative-order fractional Korteweg-de Vries equations of the formfor every α ∈ (0, 1) with homogeneous Fourier multiplier |D| −α . We tackle nonlinearities n(u) of the type |u| p or u|u| p−1 for all real p > 1, and show that when n is odd, the waves also feature antisymmetry and thus contain inverted cusps. The analysis for nonsmooth n is applicable to other negative-order nonlocal equations. Both the construction of highest antisymmetric waves and the regularisation of nonsmooth terms to an analytic bifurcation setting, we believe are new in this context.

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Section: /31mentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Periodic Hölder waves in a class of negative-order dispersive equations

Hildrum¹,

Xue²

2022

Preprint

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“…We refer to it in this work as the Whitham-Green-Naghdi (WGN) model. It has been rigorously justified among other fully dispersive models in [33,24]. 1 In this work, we numerically investigate properties of the SGN and WGN equations in extreme situations.…”

Section: Introductionmentioning

confidence: 99%

Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

Duchêne

Klein

2022

DCDS-B

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<p style='text-indent:20px;'>We perform numerical experiments on the Serre-Green-Naghdi (SGN) equations and a fully dispersive "Whitham-Green-Naghdi" (WGN) counterpart in dimension 1. In particular, solitary wave solutions of the WGN equations are constructed and their stability, along with the explicit ones of the SGN equations, is studied. Additionally, the emergence of modulated oscillations and the possibility of a blow-up of solutions in various situations is investigated.</p><p style='text-indent:20px;'>We argue that a simple numerical scheme based on a Fourier spectral method combined with the Krylov subspace iterative technique GMRES to address the elliptic problem and a fourth order explicit Runge-Kutta scheme in time allows to address efficiently even computationally challenging problems.</p>

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“…Since point 1. of the justification is already established, the long-time existence and consistency remain. Using the method of Emerald, one can prove the consistency of any Whitham-Boussinesq system with the water wave system (see also [21] for other full dispersion shallow water models). Therefore, having the long time well-posedness theory for (1.5), (1.6) and (1.7) will provide the final step for the full justification of these systems.…”

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confidence: 99%

Long time well-posedness of Whitham-Boussinesq systems

Paulsen¹

2022

Preprint

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Consideration is given to three different full dispersion Boussinesq systems arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water. We prove that, under a non-cavitation condition on the initial data, these three systems are well-posed on a time scale of order O( 1 ε ), where ε is a small parameter measuring the weak non-linearity of the waves. This result seems new for one of these systems, even for short time. The two other systems involve surface tension, and for one of them, the non-cavitation condition has to be sharpened when the surface tension is small. The proof relies on suitable symmetrizers and the classical theory of hyperbolic systems. However, we have to track the small parameters carefully in the commutator estimates to get the long time well-posedness.Finally, combining our results with the recent ones of Emerald provide a full justification of these systems as water wave models in a larger range of regimes than the classical (a, b, c, d)-Boussinesq systems.

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Rigorous Derivation from the Water Waves Equations of Some Full Dispersion Shallow Water Models

Cited by 15 publications

References 17 publications

Periodic Hölder waves in a class of negative-order dispersive equations

Periodic Hölder waves in a class of negative-order dispersive equations

Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

Long time well-posedness of Whitham-Boussinesq systems

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