arXiv subjects

Martin Oen Paulsen

Publications and source records attributed to Martin Oen Paulsen.

6 recordsLinked to original sources

arXiv2026

The shallow--water limit of linear water waves over a discontinuous bottom

We rigorously justify the zero--dispersion limit of linear water waves over a sharp bottom step, recovering the shallow-water equations as a transmission problem, and describe the localized dispersive effects generated near the discontinuity. To obtain estimates uniform in the shallow-water parameter, we study an abstract functional framework for the linear water waves equations in two-dimensional fluid domains with a Lipschitz curved bottom. The framework is based on fractional powers of the Dirichlet--Neumann operator, and we characterize low-regularity energy spaces using adapted Rellich identities. At higher regularity, the associated velocity potential develops corner singularities. For the step geometry, we use an explicit conformal mapping to obtain a version of Grisvard's shift theorem, where we derive an explicit formula for the singular part of the potential. The dependence on the horizontal variable reveals a narrow transition scale generated by the step. The convergence toward the shallow-water transmission problem holds away from this transition region, while a localized dispersive correction remains near the step. We illustrate this discrepancy numerically and identify the resulting step-induced dispersive boundary layer. Building on these observations, we consider the strongly dispersive regime and derive a consistent model that explicitly describes this boundary layer for a small step.

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arXiv2025

The F. John model and Cummins' equations for freely floating objects

In this paper, we address the well-posedness theory of F. John's problem for freely floating objects in a two-dimensional framework. This problem is a linear description of the interactions between an incompressible, irrotational free-surface fluid and a partially immersed solid object. It is related to the Cummins equations, which are a set of coupled integro-differential equations widely used by naval engineers. Our results provide a rigorous justification of this model and the first proof of its well-posedness. To this end, we show that F. John's problem has a Hamiltonian structure, albeit with a non-definite Hamiltonian which compels us to work in a semi-Hilbertian framework. The presence of corners in the fluid domain also induces a lack of elliptic regularity for the boundary value problem satisfied by the velocity potential. This is why the solution to F. John's problem has in general a limited regularity, even with very smooth initial data. Correspondingly, the solution to Cummins' equation is in general no better than $C^3$. We demonstrate higher regularity when the contact angles are small or equal to $π/2$, provided that some compatibility conditions, propagated by the flow, are satisfied. This allows us to exhibit qualitative differences between the three degrees of freedom of the object. For instance, the motion of an object allowed to move only vertically can be $C^\infty$, while it cannot be better than $C^3$ if it is only allowed to translate horizontally.

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arXiv2024

Justification of the Benjamin-Ono equation as an internal water waves model

In this paper, we give the first rigorous justification of the Benjamin-Ono equation as an internal water wave model on the physical time scale. Let $\varepsilon$ be the small parameter measuring the weak nonlinearity of the waves, $μ$ be the shallowness parameter, and $γ\in (0,1)$ is the ratio between the densities of the two fluids. To be precise, we first prove the existence of a solution to the internal water wave equations for a two-layer fluid with surface tension, where one layer is of shallow depth and the other is of infinite depth. The existence time is of order $\mathcal{O}(\frac{1}{\varepsilon})$ for a small amount of surface tension such that $\varepsilon^2 \leq\mathrm{bo}^{-1} $ where $\mathrm{bo}$ is the Bond number. Then, we show that these solutions are close, on the same time scale, to the solutions of the BO equation with a precision of order $\mathcal{O}(μ+ \mathrm{bo}^{-1})$. In addition, we provide the justification of new equations with improved dispersive properties, the Benjamin equation, and the Intermediate Long Wave (ILW) equation in the deep-water limit. The long-time well-posedness of the two-layer fluid problem was first studied by Lannes [Arch. Ration. Mech. Anal., 208(2):481-567, 2013] in the case where both fluids have finite depth. Here, we adapt this work to the case where one of the fluid domains is of finite depth, and the other one is of infinite depth. The novelties of the proof are related to the geometry of the problem, where the difference in domains alters the functional setting for the Dirichlet-Neumann operators involved. In particular, we study the various compositions of these operators that require a refined symbolic analysis of the Dirichlet-Neumann operator on infinite depth and derive new pseudo-differential estimates that might be of independent interest.

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arXiv2023

Rigorous derivation of weakly dispersive shallow water models with large amplitude topography variations

We derive rigorously from the water waves equations new irrotational shallow water models for the propagation of surface waves in the case of uneven topography in horizontal dimensions one and two. The systems are made to capture the possible change in the waves' propagation, which can occur in the case of large amplitude topography. The main contribution of this work is the construction of new multi-scale shallow water approximations of the Dirichlet-Neumann operator. We prove that the precision of these approximations is given at the order $O(μ\varepsilon)$, $O(μ\varepsilon +μ^2β^2)$ and $O(μ^2\varepsilon+μ\varepsilon β+ μ^2β^2)$. Here $μ$, $\varepsilon$, and $β$ denote respectively the shallow water parameter, the nonlinear parameter, and the bathymetry parameter. From these approximations, we derive models with the same precision as the ones above. The model with precision $O(μ\varepsilon)$ is coupled with an elliptic problem, while the other models do not present this inconvenience.

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arXiv2023

Long time well-posedness and full justification of a Whitham-Green-Naghdi system

We establish the full justification of a "Whitham-Green-Naghdi" system modeling the propagation of surface gravity waves with bathymetry in the shallow water regime. It is an asymptotic model of the water waves equations with the same dispersion relation. The model under study is a nonlocal quasilinear symmetrizable hyperbolic system without surface tension. We prove the consistency of the general water waves equations with our system at the order of precision $O(μ^2 (\varepsilon + β))$, where $μ$ is the shallow water parameter, $\varepsilon$ the nonlinearity parameter, and $β$ the topography parameter. Then we prove the long time well-posedness on a time scale $O(\frac{1}{\max\{\varepsilon,β\}})$. Lastly, we show the convergence of the solutions of the Whitham-Green-Naghdi system to the ones of the water waves equations on the later time scale.

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arXiv2022

Long time well-posedness of Whitham-Boussinesq systems

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 $\mathcal{O}(\frac1\varepsilon)$, where $\varepsilon$ 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.

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