An application of the principle of limiting absorption to the motions of floating bodies

Lenoir, M.; Martin, David W.

doi:10.1016/0022-247x(81)90032-9

Cited by 14 publications

(14 citation statements)

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“…The proof of this theorem is given in [13]. The last results suggest that in the half complex plane Im X > 0 (resp.…”

Section: Theorem 33 : For All G In Lf(fs\ P X Has a Unique Solution mentioning

confidence: 93%

“…The antisymmetry of A resulting from standard calculations, the second property alone will be proved : [13]). Then, according to the last équation of (2.10) and to property (2.3), x belongs to H 1/2 (FS).…”

Section: (24)mentioning

confidence: 96%

“…The steady-state problem Q CT + , a > 0, has been studied by F. John [1], who proved under the geometrical assumption (1.12) that provided f e if ~1 /2 (F), there is a unique solution for all values of a. Note that once the uniqueness property is known, the fact that g a + has the existence property for H~1 /2 (F) can be proved either by means of the limiting absorption principle [13] or else by means of Fredholm operators techniques (cf. Vullierme-Ledard [14]).…”

Section: Introductionmentioning

confidence: 99%

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The limiting amplitude principle applied to the motion of floating bodies

Vullierme-Ledard¹

1987

ESAIM: M2AN

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“…The proof of this theorem is given in [13]. The last results suggest that in the half complex plane Im X > 0 (resp.…”

Section: Theorem 33 : For All G In Lf(fs\ P X Has a Unique Solution mentioning

confidence: 93%

Section: (24)mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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The limiting amplitude principle applied to the motion of floating bodies

Vullierme-Ledard¹

1987

ESAIM: M2AN

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“…In general, the uniqueness results obtained via this methodology are valid for all wavelengths but have suffered, up to now, from some geometrical restrictions concerning the shape of the floating body. Having established uniqueness, the well-posedness of the problem can be deduced by using various methods, such as the integral-equation formulation [18,21,22] or the limiting absorption principle [11,24,25]. …”

mentioning

confidence: 99%

“…This simple case is free of obscured technicalities and permits us to present clearly the underlying ideas.1 Let v(6) be an arbitrary element of Lp, and consider the element 25) where the coefficients an, n = 0,1, are determined so that u(6) belongs to LPB. This is equivalent to uc(n) = 0, n = 0,1, (3.26) where uc(n) is the nth-order cosine-Fourier coefficient of the function u{6) defined…”

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confidence: 99%

Untitled

1990

Quart. Appl. Math.

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Abstract. The two-dimensional, deep-water, wave-body interaction problem for a single-hulled body, floating on the free surface of an ideal liquid, is considered. The body boundary may be nonsmooth and may intersect the free surface at arbitrary angles. The existence of a unique solution representable by a multipole-series expansion is proved for all but a discrete set of oscillation frequencies. The proof is based on the property of the associated multipoles to be a basis of the space Lp(-n,0), 1 < p < 2. Strict estimates of the form Dn -0(n~a) are also obtained for the coefficients of the multipole-series expansion for piecewise smooth (0 < a < 2) and smooth (a = 2) body boundaries.1. Introduction. Consider an infinitely long, horizontal cylinder floating on the free surface of an unbounded, infinitely deep, incompressible and inviscid liquid. In this paper we are concerned with the solvability (well-posedness) of the boundary-value problems arising when the floating body is forced, by an incident harmonic wave or by an external force, to perform time-harmonic oscillations of small amplitude about its position of stable hydrostatic equilibrium.The uniqueness question for the wave-floating body interaction problem has been studied by means of two, essentially different, methodologies. The first one is of geometric character and appropriately uses Green's theorem to ensure that the only possible solution of the homogeneous problem is the trivial one. Such an approach was inaugurated by John [18] in 1950 (see also Lenoir [24], who reworked the twodimensional case, and Kleinman [21]). Recently, Simon and Ursell [33] essentially generalized John's uniqueness theorem in a manner permitting them to consider floating and/or submerged bodies of quite a general shape. In general, the uniqueness results obtained via this methodology are valid for all wavelengths but have suffered, up to now, from some geometrical restrictions concerning the shape of the floating body. Having established uniqueness, the well-posedness of the problem can be deduced by using various methods, such as the integral-equation formulation [18,21,22] or the limiting absorption principle [11,24,25].

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An application of the limiting absorption principle to a mixed boundary value problem in an infinite strip

Doppel

Hochmuth

1995

Math Methods in App Sciences

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Communicated by E. MeisterWe present a complete proof of the existence and uniqueness of solutions of a mixed boundary value problem for the homogeneous Laplace equation in an unbounded parallel strip based on the principle of limiting absorption 1. Introduction and statement of the problem 1.1. The floating body problem is a boundary value problem concerned with determining the surface waves produced by a vertically oscillating body floating in an inviscid incompressible fluid of finite depth with a free surface extending to infinity. This problem was introduced by John [13, 141.If one wants to find approximate numerical solutions, it is a good idea to split the problem in an inner and an exterior problem (cf.[17]). The further procedure is as follows. Find the solution operator of the exterior problem and show by the boundedness of this operator, that the Neumann data of the (exterior) solution on the cut section depend continuously on the given Dirichlet data. This allows to formulate an inner problem on a bounded domain with non-local boundary conditions on the cut section, which is equivalent to the floating body problem in the infinite strip. Now there are several possibilities to discretize this localized problem, cf., for example, the localized finite element method studied by Lenoir and Tounsi [17].Because of this Lenoir and Tounsi remarked in their paper, that the existence of the solution operator for the exterior problem and its boundedness follow by the uniqueness proof given by John [14] and by the principle of limiting absorption worked out in Lenoir and Martin [16].Martin and Ursell [9]. Therefore, their argumentation to construct solutions by the limiting absorption principle breaks down. Thus, in the present paper we go back and study the existence and uniqueness of solutions of the exterior problem. 1.2. Let us denote by x = (xl, x2) the elements in the two-dimensional Euclidean space W2. Let R, := W ' x ( -h, 0) (h E W') be an undisturbed fluid domain with the bottom surface B, := R' x { -h } , the free fluid surface F , := W+ x (0) and the left boundary Co := (0) x (h, 0). The domain R, represents one-half of the exterior domain in the problem of the floating body ( cf. [2,4-7, 13, 14, 16, 171). For R E W ' we define the cut off sets RR :Then the exterior problem of the floating body problem is formulated as follows. Problem A (Classical formulation). For each x E C1(Co) and 1 E C find all u E C'(i2,) n C'(Si,) such that 0 on B , lu on F , , u = x o n & , Au = 0 in R,, a,u = where a, denotes the outer normal derivative to the domain 0,. In the case of x = 0 we call it homogeneous and inhomogeneous in the other case.1.3. We organize the paper as follows: First we put Problem A in a Hilbert space setting and show in section 2 unique solvability for 1 E @\R'. For 1 E W ' the Hilbert space methods fail. Therefore, we give a Frechet space formulation of Problem A in section 3.In the rest of the paper we prove that this Frechet space problem has, for 1 E W ' , a unique solution under the assumption...

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An application of the principle of limiting absorption to the motions of floating bodies

Cited by 14 publications

References 3 publications

The limiting amplitude principle applied to the motion of floating bodies

The limiting amplitude principle applied to the motion of floating bodies

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An application of the limiting absorption principle to a mixed boundary value problem in an infinite strip

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