Floating structures in shallow water: local well-posedness in the axisymmetric case

Bocchi, Edoardo

doi:10.48550/arxiv.1802.07643

Cited by 2 publications

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“…In Section 2 we first recall the notations for the floating structures that we have used in [3] and we write the equations for the coupled problem. Then we show that the differential equation for the solid motion can be written in a closed form by introducing an extension-trace operator, which takes the boundary value of the horizontal discharge, defined as the fluid horizontal velocity vertically integrated, in the exterior domain and gives the boundary value of the fluid height in the exterior domain.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…Finally we consider the return to equilibrium configuration, giving the initial conditions on the fluid and solid unknowns. It turns out that the compatibility conditions, which are necessary in order to apply the existence and uniqueness theorem from [3], are not satisfied for these particular initial conditions.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…This assumption is completely relevant for the situation investigated here. We showed in [3] that the floating structure problem in the case of an axisymmetric flow without swirl is described by…”

Section: ∃ H Min >mentioning

confidence: 99%

“…In our recent paper [3] we dealt with the two-dimensional (horizontal) case, we showed the local well-posedness for the axisymmetric floating structure problem in the shallow water regime for initial data regular enough, provided some compatibility conditions are satisfied. We considered a solid, with vertical sidewalls and a cylindrical symmetry, forced to move only vertically.…”

Section: Introductionmentioning

confidence: 99%

“…The floating structure problem for a viscous fluid in a one dimensional bounded domain is considered in [11]. The aim of this paper is to extend the work of Lannes on the return to equilibrium problem to the two-dimensional case taking into account nonlinear effects and using the same framework as in [3], which means that we consider here the axisymmetric setting, the shallow water approximation for the fluid and a solid with the properties we have stated before. An important change with respect to the one-dimensional case is the presence of delay terms in the equation governing the solid motion.…”

Section: Introductionmentioning

confidence: 99%

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On the return to equilibrium problem for axisymmetric floating structures in shallow water

Bocchi

2019

Preprint

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In this paper we address the return to equilibrium problem for an axisymmetric floating structure in shallow water. First we show that the motion of the solid object can be reduced to a delay differential equation involving an extension-trace operator whose role is to describe the influence of the fluid equations on the solid motion. It turns out that the compatibility conditions on the initial data for the return to equilibrium configuration are not satisfied, so we cannot use the results from [3] for the nonlinear problem. Hence we linearize the equations in the exterior domain supposing small amplitude waves and we keep the nonlinear equations in the interior domain. For such configurations, the extension-trace operator can be computed explicitly and the delay term in the delay differential equation can be put in convolution form. The solid motion is governed by a nonlinear second order integro-differential equation, whose linearization is the well-known Cummins equation. By writing it as a functional differential equation with infinite delay, we show the global in time existence and uniqueness of the solution using the conservation of the total energy. Finally the local asymptotic stability of the equilibrium position is shown.

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Section: Outline Of the Papermentioning

confidence: 99%

Section: Outline Of the Papermentioning

confidence: 99%

Section: ∃ H Min >mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the return to equilibrium problem for axisymmetric floating structures in shallow water

Bocchi

2019

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Hyperbolic free boundary problems and applications to wave-structure interactions

Iguchi,

Lannes

2018

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Motivated by a new kind of initial boundary value problem (IBVP) with a free boundary arising in wave-structure interaction, we propose here a general approach to onedimensional IBVP as well as transmission problems. For general strictly hyperbolic 2 × 2 quasilinear hyperbolic systems, we derive new sharp linear estimates with refined dependence on the source term and control on the traces of the solution at the boundary. These new estimates are used to obtain sharp results for quasilinear IBVP and transmission problems, and for fixed, moving, and free boundaries. In the latter case, two kinds of evolution equations are considered. The first one is of "kinematic type" in the sense that the velocity of the interface has the same regularity as the trace of the solution. Several applications that fall into this category are considered: the interaction of waves with a lateral piston, and a new version of the well-known stability of shocks (classical and undercompressive) that improves the results of the general theory by taking advantage of the specificities of the one-dimensional case. We also consider "fully nonlinear" evolution equations characterized by the fact that the velocity of the interface is one derivative more singular than the trace of the solution. This configuration is the most challenging; it is motivated by a free boundary problem arising in wave-structure interaction, namely, the evolution of the contact line between a floating object and the water. This problem is solved as an application of the general theory developed here.

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Floating structures in shallow water: local well-posedness in the axisymmetric case

Cited by 2 publications

References 0 publications

On the return to equilibrium problem for axisymmetric floating structures in shallow water

On the return to equilibrium problem for axisymmetric floating structures in shallow water

Hyperbolic free boundary problems and applications to wave-structure interactions

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