Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On the solvability of a two-dimensional water-wave radiation problem


Author: G. A. Athanassoulis
Journal: Quart. Appl. Math. 44 (1987), 601-620
MSC: Primary 76B15; Secondary 35R35
DOI: https://doi.org/10.1090/qam/872813
MathSciNet review: 872813
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Abstract: The existence of a unique weak solution for the two-dimensional water-wave radiation problem arising when a floating rigid body oscillates on the free surface is established for all but a discrete set of oscillation frequencies. The body boundary is taken to be of $ C_ * ^{1,\alpha }$ class (see Sec. 2) and the body boundary condition is satisfied in the $ {L^2}$-sense. The proof relies on an expansion theorem (Athanassoulis [1]) and on the property of the associated water-wave multipoles being a Riesz basis of $ {L^2}\left( { - \pi ,0} \right)$, a fact which is established in the present paper. Under stronger geometrical restrictions on the body boundary it is proved, using a method due to Ursell [10], that the weak solution is actually a classical one; that is, the velocity field is continuous up to ana including the body boundary.


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DOI: https://doi.org/10.1090/qam/872813
Article copyright: © Copyright 1987 American Mathematical Society


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