Long Time Existence for a Strongly Dispersive Boussinesq System

Saut, Jean‐Claude; Xu, Li

doi:10.1137/19m1250698

Cited by 12 publications

(7 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many strategies exist to study the water-wave problem especially by deriving equivalent models with better mathematical structure such as well-posedness, conservation of energy, solitary waves, or physical properties (see for instance [2,3,[5][6][7]28,30,34,[36][37][38][39][40]). It is worth noticing that the well posed results for such model exist on a time scale of order 1/ √ ε (methods based on dispersive estimate in [44]) and 1/ε (energy estimate method in [28] ).…”

Section: The Water-wave Equationsmentioning

confidence: 99%

Mathematical modeling and numerical analysis for the higher order Boussinesq system

et al. 2022

View full text Add to dashboard Cite

This study deals with higher-ordered asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order 1/ [[EQUATION]] and showing that the solution's behavior is close to the solution of the water waves equations with a better precision corresponding to initial data, the asymptotic model is well-posed in the sense of Hadamard. Then we compared several water waves solitary solutions with respect to the numerical solution of our model. At last, we solve explicitly this model and validate the results numerically.

show abstract

Section: The Water-wave Equationsmentioning

confidence: 99%

Mathematical modeling and numerical analysis for the higher order Boussinesq system

et al. 2022

View full text Add to dashboard Cite

show abstract

“…These models are second order partial differential equations with no higher order derivative terms, and the initial-value problem is well-posed for all of the systems in the hierarchy. The only disadvantage is that the Green-Naghdi/Boussinesq systems [9,8,6,20,5,4,3,26,25,21] are not among the models in this hierarchy whose order in µ varies with bottom topographies.…”

Section: Introductionmentioning

confidence: 99%

Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations

Khorbatly

2023

DCDS-B

View full text Add to dashboard Cite

<p style='text-indent:20px;'>In the mathematical theory of water waves, this paper focuses on the hierarchy of higher order asymptotic models. The well-posedness of the medium amplitude extended Green-Naghdi model, as well as higher-ordered Boussinesq-Peregrine and Boussinesq models, is first demonstrated. Introducing a regularization term and various physical topography variations, we show that these models admit unique solutions by a standard energy estimate method in the "hyperbolic" space <inline-formula><tex-math id="M1">\begin{document}$ H^{s+2}( \mathbb R)^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ s>3/2 $\end{document}</tex-math></inline-formula>, but on short/intermediate time scales with respect to amplitude and topography parameters of order <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon^{-1/4} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \max(\sqrt{ \varepsilon}, \beta)^{-1} = \beta^{-1} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \varepsilon^{-1/2} $\end{document}</tex-math></inline-formula> respectively. Furthermore, we show that the extended Green-Naghdi system's long-term well-posedness is reachable on time scales of order <inline-formula><tex-math id="M6">\begin{document}$ \max( \varepsilon, \beta)^{-1} $\end{document}</tex-math></inline-formula>. The above three specified models, in particular, admit longer time existence of order <inline-formula><tex-math id="M7">\begin{document}$ \varepsilon^{-1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \max( \varepsilon, \beta)^{-1} $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ \varepsilon^{-1} $\end{document}</tex-math></inline-formula>, respectively.</p>

show abstract

“…There has been several studies by J-C. Saut et al [6,9,10,11,8] (see also [7]) regarding long-time existence of solutions for Boussinesq type equations with initial data of size O (1) in some Sobolev norm, where the time of existence depends on shallowness parameter ǫ. In [10,11,8] the analysis is based only on symmetrization and energy techniques, and do not exploit the dispersive properties of the equations.…”

Section: Introductionmentioning

confidence: 99%

“…The time of existence obtained for the equations involved is at most of scale O (1/ǫ) but the space of resolutions are smaller. On the other hand, in [6] and [9] the dispersive nature of the systems involved is used to study the long-time existence problem. For instance, in [6] using the dispersive method the authors proved that the two dimensional dispersive Boussinesq system of the form…”

Section: Introductionmentioning

confidence: 99%

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

Tesfahun¹

2022

Preprint

View full text Add to dashboard Cite

This paper is concerned with a two dimensional Whitham-Boussinesq system modeling surface waves of an inviscid incompressible fluid layer. We prove that the associated Cauchy problem is well-posed for initial data of low regularity, with existence time of scale O 1/ √ ǫ , where ǫ > 0 is a shallowness parameter measuring the ratio of the amplitude of the wave to the mean depth of the fluid. The key ingredients in the proof are frequency loacalised dispersive and Strichartz estimates that depend on ǫ as well as bilinear estimates in some Strichartz norms.

show abstract

Long Time Existence for a Strongly Dispersive Boussinesq System

Cited by 12 publications

References 31 publications

Mathematical modeling and numerical analysis for the higher order Boussinesq system

Mathematical modeling and numerical analysis for the higher order Boussinesq system

Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations

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

Contact Info

Product

Resources

About