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On the Linear Water Wave Problem in the Presence of a Critically Submerged Body
UCL, London, England .
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
2012 (English)In: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7145, Vol. 44, no 6, 4222-4249 p.Article in journal (Refereed) Published
Abstract [en]

We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a critically submerged body (i.e., the body touching the water surface). Assuming uniqueness of the solution in the energy space, we prove the existence of a solution which satisfies the radiation conditions at infinity as well as at the cusp point where the body touches the water surface. This solution is obtained by the limiting absorption procedure. Next we introduce a relevant scattering matrix and analyze its properties. Under a geometric condition introduced by V. Mazya in 1978, we prove an important property of the scattering matrix, which may be interpreted as the absence of total internal reflection. This property also allows us to obtain uniqueness and existence of a solution in some function spaces (e.g., H-loc(2) boolean AND L-infinity) without use of the radiation conditions and the limiting absorption principle, provided a spectral parameter in the boundary conditions on the surface of the water is large enough. The fact that the existence and uniqueness result does not rely on either the radiation conditions or the limiting absorption principle is the first result of this type known to us in the theory of linear wave problems in unbounded domains.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics , 2012. Vol. 44, no 6, 4222-4249 p.
Keyword [en]
water waves, limiting absorption principle, radiation conditions, uniqueness, domains with cusps
National Category
Natural Sciences
URN: urn:nbn:se:liu:diva-87969DOI: 10.1137/120868074ISI: 000312727800020OAI: diva2:601022
Available from: 2013-01-28 Created: 2013-01-28 Last updated: 2013-03-22

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