Steady water waves

Strauss, Walter A.

doi:10.1090/s0273-0979-2010-01302-1

Cited by 81 publications

(61 citation statements)

References 83 publications

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“…However, the term we have identified as making the primary contribution to ill-posedness is (Λξ ) 2 . In a naive energy estimate, this first-derivative term would throw up a quarter of a derivative more than can be controlled by the Kato smoothing.…”

Section: Discussionmentioning

confidence: 99%

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On ill-posedness of truncated series models for water waves

2014

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The evolution of surface gravity waves on a large body of water, such as an ocean, is reasonably well approximated by the Euler system for ideal, freesurface flow under the influence of gravity. The well-posedness theory for initial-value problems for these equations, which has a long and distinguished history, reveals that solutions exist, are unique, and depend continuously upon initial data in various function-space contexts. This theory is subtle, and the design of stable, accurate, numerical schemes is likewise challenging. Depending upon the wave regime in question, there are many different approximate models that can be formally derived from the Euler equations. As the Euler system is known to be well-posed, it seems appropriate that associated approximate models should also have this property. This study is directed to this issue. Evidence is presented calling into question the well-posedness of a well-known class of model equations which are widely used in simulations. A simplified version of these models is shown explicitly to be ill-posed and numerical simulations of quadratic-and cubicorder water-wave models, initiated with initial data predicted by the explicit analysis of the simplified model, lends credence to the general contention that these models are ill-posed.

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Section: Discussionmentioning

confidence: 99%

“…To be more precise, the term (Λξ ) 2 , which can be viewed as a parabolic term of indefinite sign, appears to be the cause of the trouble. Incidentally, we are not proposing (2.3) as a new model of water-waves, but are simply using it to help understand the dynamics of solutions of (2.1) and (2.2).…”

Section: Water Wave Modelsmentioning

confidence: 99%

On ill-posedness of truncated series models for water waves

2014

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“…Consider a two-dimensional region filled with an ideal fluid, that is, an incompressible fluid (of constant density) without viscosity. Although an ideal fluid is a mathematical abstraction (since all real fluids are at least very weakly viscous), the concept is useful in certain problems because fluids having small viscosities (for example water [21,30]) behave as if they are inviscid; close to absolute zero temperature, helium is practically an ideal fluid [19], and the two-dimensional ideal fluid setting is appropriate for certain geophysical flows [3,16].…”

Section: The Equations Of Motionmentioning

confidence: 99%

Harmonic Maps and Ideal Fluid Flows

Aleman

Constantin

2011

Arch Rational Mech Anal

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Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations. More precisely, the problem of finding all solutions which in Lagrangian variables (describing the particle paths of the flow) present a labelling by harmonic functions is reduced to solving an explicit nonlinear differential system in C n with n = 3 or n = 4. While the general solution is not available in explicit form, structural properties of the system permit us to identify several families of explicit solutions.

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“…Nearly 80 years after Stokes' work, the method for finite water depth was extended to the third order by Borgman & Chappelear [15] and to the fifth order by Skjelbreia & Hendrickson [16] and Fenton [17]. In the last two decades, the power series approach has been replaced by a global bifurcation approach in the analysis of waves of large amplitude [18]. For practical purposes, Sainflou [19] derived a nonlinear standing-wave equation based on the trochoidal wave theory, and later Miche [20] obtained the second-order equation for zero mass transport.…”

Section: Introductionmentioning

confidence: 99%

Eulerian–Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry

Umeyama

2012

Phil. Trans. R. Soc. A.

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This paper investigates the velocity and the trajectory of water particles under surface waves, which propagate at a constant water depth, using particle image velocimetry (PIV). The vector fields and vertical distributions of velocities are presented at several phases in one wave cycle. The third-order Stokes wave theory was employed to express the physical quantities. The PIV technique's ability to measure both temporal and spatial variations of the velocity was proved after a series of attempts. This technique was applied to the prediction of particle trajectory in an Eulerian scheme. Furthermore, the measured particle path was compared with the positions found theoretically by integrating the Eulerian velocity to the higher order of a Taylor series expansion. The profile of average travelling distance is also presented with a solution of zero net mass flux in a closed wave flume.

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Steady water waves

Abstract: Abstract. We present a survey of certain aspects of the theory of steady water waves with emphasis on the role played by vorticity. Historical background, numerical illustrations, and brief discussions of the time-dependent problem and of approximate models are included as well.

Cited by 81 publications

References 83 publications

On ill-posedness of truncated series models for water waves

On ill-posedness of truncated series models for water waves

Harmonic Maps and Ideal Fluid Flows

Eulerian–Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry

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