Local well-posedness of quasi-linear systems generalizing KdV

Akhunov, Timur

doi:10.3934/cpaa.2013.12.899

Cited by 15 publications

(35 citation statements)

References 22 publications

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Section: Discussionmentioning

confidence: 99%

“…A study of the competition between dispersion and antidiffusion that the first author and Wright have carried out for KdV-like equations in [12] (and see also Akhunov [13]) gives some clues to the present situation. It is found in references [12,13] that the Kato smoothing effect from the dispersive terms must be strong enough to counteract the growth inherent in the anti-diffusion. In the shallow-water KdV case, there is a loss of one derivative from the anti-diffusion (the anti-diffusion comes from a second derivative term of indeterminate sign, leading to terms in the energy estimates with one more derivative than can be controlled through a naive estimate), but the Kato smoothing effect from the leading-order, dispersive term is also one derivative.…”

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%

Section: Discussionmentioning

confidence: 99%

On ill-posedness of truncated series models for water waves

2014

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“…We then apply the semigroup estimate (2.6) to obtain Applying identical estimates for the difference T [W (1) ] − T [W (2) ] we see that we may choose the timescale 0 < T = T (ν) ≪ 1 sufficiently small so that the map T : B → B is a contraction on B. The result then follows from an application of the contraction principle.…”

Section: 2mentioning

confidence: 94%

“…We then have the following estimate for the linearized equation: Proof. Given any two solutions W (1) , W (2) of (3.7) we define (2) .…”

Section: 5mentioning

confidence: 99%

Existence and Uniqueness of Solutions for a Quasilinear KdV Equation with Degenerate Dispersion

Germain

Harrop-Griffiths

Marzuola

2019

Comm Pure Appl Math

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We consider a quasilinear KdV equation that admits compactly supported traveling wave solutions (compactons). This model is one of the most straightforward instances of degenerate dispersion, a phenomenon that appears in a variety of physical settings as diverse as sedimentation, magma dynamics and shallow water waves. We prove the existence and uniqueness of solutions with sufficiently smooth, spatially localized initial data. © 2019 Wiley Periodicals, Inc.

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“…The proof of the proposition follow the outline of a standard semilinear parabolic problem, e.g. [Akh13]. The linear semi-group gains 3 derivatives in L 2 -based spaces and allows the f (∂ 3…”

Section: Regularizationsmentioning

confidence: 99%

Well-posedness of fully nonlinear KdV-type evolution equations

Akhunov¹,

Ambrose²,

Wright³

2019

Nonlinearity

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We study the well-posedness of the initial value problem for fully nonlinear evolution equations, ut = f [u], where f may depend on up to the first three spatial derivatives of u. We make three primary assumptions about the form of f : a regularity assumption, a dispersivity assumption, and an assumption related to the strength of backwards diffusion. Because the third derivative of u is present in the right-hand side and we effectively assume that the equation is dispersive, we say that these fully nonlinear evolution equations are of KdV-type. We prove the well-posedness of the initial value problem in the Sobolev space H 7 (R). The proof relies on gauged energy estimates which follow after making two regularizations, a parabolic regularization and mollification of the initial data.

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Local well-posedness of quasi-linear systems generalizing KdV

Cited by 15 publications

References 22 publications

On ill-posedness of truncated series models for water waves

On ill-posedness of truncated series models for water waves

Existence and Uniqueness of Solutions for a Quasilinear KdV Equation with Degenerate Dispersion

Well-posedness of fully nonlinear KdV-type evolution equations

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