Recent advances on the global regularity for irrotational water waves

Ionescu, Alexandru D.; Pusateri, Fabio

doi:10.1098/rsta.2017.0089

Cited by 24 publications

(22 citation statements)

References 161 publications

(346 reference statements)

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“…the generalization to non flat bottoms is given in (96). Note that contrary to what happens for the horizontal velocity, the contribution of the vorticity to the vertical velocity is smaller than the irrotational contribution (and this remains true for larger vorticities).…”

Section: 2mentioning

confidence: 96%

Modeling shallow water waves

Lannes

2020

Nonlinearity

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We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the "turbulent" and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre-Green-Naghdi (SGN) equations for which we review several pending open problems. More recent models such as the multi-layer NSW or SGN systems, as well as the Isobe-Kakinuma equations are also reviewed under a unified formalism that should simplify comparisons. We also comment on the scalar versions of the various shallow water systems which can be used to describe unidirectional waves in horizontal dimension d = 1; among them are the KdV, BBM, Camassa-Holm and Whitham equations. Finally, we show how to take vorticity effects into account in shallow water modeling, with specific focus on the behavior of the turbulent terms. As examples of challenges that go beyond the present scope of mathematical justification, we review recent works using shallow water models with vorticity to describe wave breaking, and also derive models for the propagation of shallow water waves over strong currents.

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Section: 2mentioning

confidence: 96%

Modeling shallow water waves

Lannes

2020

Nonlinearity

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“…Some of the above infinite-depth results have been extended to hold in the context of flat geometry (e.g., see [102,103]). The reader wishing to learn more about the global regularity problem for the water waves system can consult the survey [70], which contains an enlightening summary of the recent progress on the global regularity problem for water waves and includes an extensive list of references.…”

Section: A Brief History Of the Water Waves Problemmentioning

confidence: 99%

“…where e 1 is the remainder. We now examine d 2 , again plugging in for γ δ t and γ δ t from (70). These substitutions yield…”

Section: Existence Of Solutionsmentioning

confidence: 99%

Local Well-Posedness of the Gravity-Capillary Water Waves System in the Presence of Geometry and Damping

Moon¹

2022

Preprint

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We consider the gravity-capillary water waves problem in a domain Ωt Ă T ˆR with substantial geometric features. Namely, we consider a variable bottom, smooth obstacles in the flow and a constant background current. We utilize a vortex sheet model introduced by Ambrose, et. al. in [16], which is an extension of the vortex sheet model studied in [14,17]. We show that the water waves problem is locally-in-time well-posed in this geometric setting and study the lifespan of solutions. We then add a damping term and derive evolution equations that account for the damper. Ultimately, we show that the same well-posedness and lifespan results apply to the damped system. We primarily utilize energy methods; particularly our approach here closely follows the approach taken in [14].

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“…-Since the governing equations for water waves are highly nonlinear, solutions that describe realistic fluid motions are elusive and the development of singularities in classical solutions (in the form of wave breaking) is one of the most difficult unanswered questions. However, on the related issue of long-time existence of solutions of small amplitude there has recently been significant progress, discussed in [50]. -The paper [51] is a broadly informative review of the history and present research directions associated with the phenomenon of Stokes drift by surface gravity waves.…”

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confidence: 99%

“…However, on the related issue of long-time existence of solutions of small amplitude there has recently been significant progress, discussed in [50]. -The paper [51] is a broadly informative review of the history and present research directions associated with the phenomenon of Stokes drift by surface gravity waves.…”

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confidence: 99%

Nonlinear water waves: introduction and overview

Constantin

2017

Phil. Trans. R. Soc. A.

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For more than two centuries progress in the study of water waves proved to be interdependent with innovative and deep developments in theoretical and experimental directions of investigation. In recent years, considerable progress has been achieved towards the understanding of waves of large amplitude. Within this setting one cannot rely on linear theory as nonlinearity becomes an essential feature. Various analytic methods have been developed and adapted to come to terms with the challenges encountered in settings where approximations (such as those provided by linear or weakly nonlinear theory) are ineffective. Without relying on simpler models, progress becomes contingent upon the discovery of structural properties, the exploitation of which requires a combination of creative ideas and state-of-the-art technical tools. The successful quest for structure often reveals unexpected patterns and confers aesthetic value on some of these studies. The topics covered in this issue are both multi-disciplinary and interdisciplinary: there is a strong interplay between mathematical analysis, numerical computation and experimental/field data, interacting with each other via mutual stimulation and feedback. This theme issue reflects some of the new important developments that were discussed during the programme ‘Nonlinear water waves’ that took place at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) from 31st July to 25th August 2017. A cross-section of the experts in the study of water waves who participated in the programme authored the collected papers. These papers illustrate the diversity, intensity and interconnectivity of the current research activity in this area. They offer new insight, present emerging theoretical methodologies and computational approaches, and describe sophisticated experimental results. This article is part of the theme issue ‘Nonlinear water waves’.

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Recent advances on the global regularity for irrotational water waves

Cited by 24 publications

References 161 publications

Modeling shallow water waves

Modeling shallow water waves

Local Well-Posedness of the Gravity-Capillary Water Waves System in the Presence of Geometry and Damping

Nonlinear water waves: introduction and overview

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