Long-time existence for a Whitham--Boussinesq system in two dimensions

Tesfahun, Achenef

doi:10.48550/arxiv.2201.03628

Cited by 2 publications

(2 citation statements)

References 8 publications

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“…The estimates derived in the aforementioned papers are not uniform in μ. However, a recent study by Tesfahun [49] proved that the corresponding two-dimensional system (1.10) without surface tension is well-posed on a time interval of order O( 1 √ ε ) in the KdV-regime. Indeed, dispersive techniques are tailored-made for short waves and therefore seem not to be well suited to capture the long wave regime (see for instance [35] for similar results for the Boussinesq system in the KdV-KdV case).…”

Section: Former Well-posedness Resultsmentioning

confidence: 99%

Long time well-posedness of Whitham–Boussinesq systems

Paulsen

2022

Nonlinearity

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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 ɛ is a small parameter measuring the weak nonlinearity of the waves. For one of the systems, this result is new 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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Section: Former Well-posedness Resultsmentioning

confidence: 99%

Long time well-posedness of Whitham–Boussinesq systems

Paulsen

2022

Nonlinearity

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“…The estimates derived in the aforementioned papers are not uniform 4 in µ. However, a recent study by Tesfahun [49] proved that the system corresponding to (1.7) in the 2-dimensional case and without surface tension is well-posed on a time interval of order O( 1 √ ε ) in the KdV−regime. Indeed, dispersive techniques are tailored-made for short waves and therefore seem not to be well suited to capture the long wave regime (see for instance [35] for similar results for the Boussinesq system in the KdV-KdV case).…”

mentioning

confidence: 96%

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.

show abstract

Long-time existence for a Whitham--Boussinesq system in two dimensions

Cited by 2 publications

References 8 publications

Long time well-posedness of Whitham–Boussinesq systems

Long time well-posedness of Whitham–Boussinesq systems

Long time well-posedness of Whitham-Boussinesq systems

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