A Note on the Local Well-Posedness for the Whitham Equation

Ehrnström, Mats; Escher, Joachim; Pei, Long

doi:10.1007/978-3-319-12547-3_3

Cited by 31 publications

(46 citation statements)

References 31 publications

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“…0.35. It is known from Ehrnström, Pei, Wang [14] that we are in a locally well posed situation, however, the obtained solution seems very unstable as one can see in Figure 8. This experiment was repeated with different time integrators, including the symplectic first-order Euler method described in Section 4.…”

Section: Numerical Experimentsmentioning

confidence: 83%

A comparative study of bi-directional Whitham systems

Dinvay

Dutykh

Kalisch

2019

Applied Numerical Mathematics

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In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties.In the current work, we review some of the existing fully dispersive systems, such as found in [1,3,8,17,23,24]. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

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Section: Numerical Experimentsmentioning

confidence: 83%

A comparative study of bi-directional Whitham systems

Dinvay

Dutykh

Kalisch

2019

Applied Numerical Mathematics

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“…The BDW system was formally derived in [1,21] from the incompressible Euler equations to model fully dispersive shallow water waves whose propagation is allowed to be both left-and rightward, and appeared in [19,22] as a full dispersion system in the Boussinesq regime with the dispersion of the water waves system. There have been several investigations on the BDW system: local well-posedness [13,18] (in homogeneous Sobolev spaces at a positive background), a logarithmically cusped wave of greatest height [11]. There are also numerical results, investigating the validity of the BDW system as a model of waves on shallow water [4], numerical bifurcation and spectral stability [5] and the observation of dispersive shock waves [24].…”

Section: Introductionmentioning

confidence: 99%

Solitary wave solutions to a class of Whitham–Boussinesq systems

Nilsson

Wang

2019

Z. Angew. Math. Phys.

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In this note we study solitary wave solutions of a class of Whitham-Boussinesq systems which includes the bi-directional Whitham system as a special example. The travelling wave version of the evolution system can be reduced to a single evolution equation, similar to a class of equations studied by Ehrnström, Groves and Wahlén [10]. In that paper the authors prove the existence of solitary wave solutions using a constrained minimization argument adapted to noncoercive functionals, developed by Buffoni [3], Groves and Wahlén [15], together with the concentration-compactness principle.2010 Mathematics Subject Classification. 76B15; 76B25, 35S30, 35A20.

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“…Moreover, if l above is sufficiently negative then the dispersion of L is very weak, and global well-posedness of (1.1) fails in classical energy spaces. In fact, the Whitham equation as a typical representative of (1.1) is locally well-posed in Sobolev spaces H s , s > 3/2, with localized or periodic initial data [9,20], but exhibits finite-time blow-up (wave-breaking) for some sufficiently smooth initial data [7,13,23] so that global well-posedness in H s (R), s > 3/2, is not possible. This kind of break-up phenomena cannot be observed in equations with strong dispersion like KdV and BO, which are globally well-posed in H s (R) for all s ≥ 1 (see [6] and [27], respectively).…”

Section: Introductionmentioning

confidence: 99%

Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

Ehrnström

Pei

2018

J. Evol. Equ.

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Abstract. For both localized and periodic initial data, we prove local existence in classical energy space H s , s > 3 2 , for a class of dispersive equations u t +(n(u)) x + Lu x = 0 with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators L whose symbol is of temperate growth, and n(·) in the local Sobolev space H s+2 loc (R). In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semigroup methods and a new composition result for Besov spaces. In particular, we extend a previous result for Nemytskii operators on Besov spaces on R to the periodic setting by using the difference-derivative characterization of Besov spaces.

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A Note on the Local Well-Posedness for the Whitham Equation

Cited by 31 publications

References 31 publications

A comparative study of bi-directional Whitham systems

A comparative study of bi-directional Whitham systems

Solitary wave solutions to a class of Whitham–Boussinesq systems

Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

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