Radiation conditions at the top of a rotational cusp in the theory of water-waves
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 5, p. 947-979

We study the linearized water-wave problem in a bounded domain (e.g. a finite pond of water) of ${ℝ}^{3}$, having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered that in this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point $𝒪$ of the water surface, where a submerged body touches the surface (see Fig. 1). The radiation conditions emerge from the requirement that the linear operator associated to the problem be Fredholm of index zero in relevant weighted function spaces with separated asymptotics. The classification of incoming and outgoing (seen from $𝒪$) waves and the unitary scattering matrix are introduced.

DOI : https://doi.org/10.1051/m2an/2011004
Classification:  76B15,  35J25,  35P99
Keywords: linear water-wave problem, cuspidal domain, radiation condition, scattering matrix
@article{M2AN_2011__45_5_947_0,
author = {Nazarov, Sergey A. and Taskinen, Jari},
title = {Radiation conditions at the top of a rotational cusp in the theory of water-waves},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {45},
number = {5},
year = {2011},
pages = {947-979},
doi = {10.1051/m2an/2011004},
zbl = {1267.76013},
mrnumber = {2817552},
language = {en},
url = {http://www.numdam.org/item/M2AN_2011__45_5_947_0}
}

Nazarov, Sergey A.; Taskinen, Jari. Radiation conditions at the top of a rotational cusp in the theory of water-waves. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 5, pp. 947-979. doi : 10.1051/m2an/2011004. http://www.numdam.org/item/M2AN_2011__45_5_947_0/

[1] F.L. Bakharev and S.A. Nazarov, On the structure of the spectrum of the elasticity problem for a body with a super-sharp spike. Sibirsk. Mat. Zh. 50 (2009) 746-756. (English transl. Siberian Math. J. 50 (2009).) | MR 2583612 | Zbl 1224.35395

[2] M.S. Birman and M.Z. Solomyak, Spectral theory of self-adjoint operators in Hilber space. Reidel Publ. Company, Dordrecht (1986). | Zbl 0744.47017

[3] A.-S. Bonnet-Ben Dhia, P. Joly, Mathematical analysis of guided water waves. SIAM J. Appl. Math. 53 (1993). | MR 1247167 | Zbl 0787.76007

[4] G. Cardone, S.A. Nazarov and J. Sokolowski, Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary. Asymptotic Analysis 62 (2009) 41-88. | MR 2514663 | Zbl 1175.35040

[5] G. Cardone, S.A. Nazarov and J. Taskinen, The “absorption" effect caused by beak-shaped boundary irregularity for elastic waves. Dokl. Ross. Akad. Nauk. 425 (2009) 182-186. (English transl. Doklady Physics 54 (2009) 146-150.) | MR 2537138 | Zbl pre05620089

[6] G. Cardone, S.A. Nazarov and J. Taskinen, Criteria for the existence of the essential spectrum for beak-shaped elastic bodies. J. Math. Pures Appl. (to appear) | Zbl pre05653511

[7] D. Daners, Robin boundary value problems on arbitrary domains. Trans. Amer. Math. Soc. 352 (2000) 4207-4236. | MR 1650081 | Zbl 0947.35072

[8] D. Daners, A Faber-Krahn inequality for Robin problems in any space dimension. Math. Annal. 335 (2006) 767-785. | MR 2232016 | Zbl 1220.35103

[9] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Die Grundlehren der mathematischen Wissenschaften 224. Springer, Berlin (1977). | MR 473443 | Zbl 0361.35003

[10] D.S. Jones, The eigenvalues of ${\nabla }^{2}u+\lambda u=0$ when the boundary conditions are given on semi-infinite domains. Proc. Camb. Phil. Soc. 49 (1953) 668-684. | MR 58086 | Zbl 0051.07704

[11] V.A. Kondratiev, Boundary value problems for elliptic problems in domains with conical or corner points. Trudy Moskov. Mat. Obshch. 16 (1967) 209-292. (English transl. Trans. Moscow Mat. Soc. 16 (1967) 227-313.) | MR 226187 | Zbl 0194.13405

[12] V.A. Kozlov, V.G. Maz'Ya and J. Rossmann, Elliptic boundary value problems in domains with point singularities. Mathematical Surveys and Monographs 52. American Mathematical Society, Providence, RI (1997). | MR 1469972 | Zbl 0947.35004

[13] V.A. Kozlov, V.G. Maz'Ya and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations. Mathematical Surveys and Monographs 85. American Mathematical Society, Providence, RI (2001). | MR 1788991 | Zbl 0965.35003

[14] V.V. Krylov, New type of vibration dampers utilising the effect of acoustic “black holes". Acta Acustica united with Acustica 90 (2004) 830-837.

[15] N. Kuznetsov, V. Maz'Ya and B. Vainberg, Linear Water Waves. Cambridge University Press, Cambridge (2002). | MR 1925354 | Zbl 0996.76001

[16] O.A. Ladyzhenskaya, Boundary value problems of mathematical physics. Springer Verlag, New York (1985). | MR 793735 | Zbl 0588.35003

[17] J.L. Lions and E. Magenes, Non-homogeneus boundary value problems and applications (French). Dunod, Paris (1968). (English transl. Springer-Verlag, Berlin-Heidelberg-New York (1972).) | Zbl 0223.35039

[18] V. Mazya, Sobolev spaces, translated from the Russian by T.O. Shaposhnikova. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin (1985). | MR 817985 | Zbl 1217.46002

[19] V.G. Maz'Ya and B.A. Plamenevskii, The asymptotic behavior of solutions of differential equations in Hilbert space. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972) 1080-1133; erratum, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 709-710. | MR 352728 | Zbl 0266.34067

[20] V.G, Mazja and B.A. Plamenevskii, On coefficients in asymptotics of solutions of elliptic boundary value problems in a domain with conical points. Math. Nachr. 76 (1977) 29-60. (Engl. transl. Amer. Math. Soc. Transl. 123 (1984) 57-89.) | MR 601608 | Zbl 0554.35036

[21] V.G. Mazja and B.A. Plamenevskii, Estimates in Lp and Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. Math. Nachr. 81 (1978) 25-82. (Engl. transl. Amer. Math. Soc. Transl. (Ser. 2) 123 (1984) 1-56 .) | MR 492821 | Zbl 0554.35035

[22] V.G. Mazya and S.V. Poborchi, Imbedding and Extension Theorems for Functions on Non-Lipschitz Domains. SPbGU publishing (2006).

[23] M.A. Mironov, Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval. Soviet Physics-Acoustics 34 (1988) 318-319.

[24] S.A. Nazarov Asymptotics of the solution of the Neumann problem at a point of tangency of smooth components of the boundary of the domain. Izv. Ross. Akad. Nauk. Ser. Mat. 58 (1994) 92-120. (English transl. Math. Izvestiya 44 (1995) 91-118.) | MR 1271516 | Zbl 0841.35030

[25] S.A. Nazarov, On the flow of water under a still stone. Mat. Sbornik 186 (1995) 75-110. (English transl. Math. Sbornik 186 (1995) 1621-1658.) | MR 1368787 | Zbl 0863.35078

[26] S.A. Nazarov, A general scheme for averaging self-adjoint elliptic systems in multidimensional domains, including thin domains. Algebra Analiz. 7 (1995) 1-92. (English transl. St. Petersburg Math. J. 7 (1996) 681-748.) | MR 1365812 | Zbl 0859.35025

[27] S.A. Nazarov, The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes. Uspehi Mat. Nauk. 54 (1999) 77-142. (English transl. Russ. Math. Surveys. 54 (1999) 947-1014.) | MR 1741662 | Zbl 0970.35026

[28] S.A. Nazarov, Weighted spaces with detached asymptotics in application to the Navier-Stokes equations. in: Advances in Mathematical Fluid Mechanics. Paseky, Czech. Republic (1999) 159-191. Springer-Verlag, Berlin (2000). | MR 1863212 | Zbl 0974.35088

[29] S.A. Nazarov, The Navier-Stokes problem in a two-dimensional domain with angular outlets to infinity. Zap. Nauchn. Sem. St.-Petersburg Otdel. Mat. Inst. Steklov 257 (1999) 207-227. (English transl. J. Math. Sci. 108 (2002) 790-805.) | MR 1754702 | Zbl 0979.35115

[30] S.A. Nazarov, The spectrum of the elasticity problem for a spiked body. Sibirsk. Mat. Zh. 49 (2008) 1105-1127. (English transl. Siberian Math. J. 49 (2008) 874-893.) | MR 2469058 | Zbl 1224.35397

[31] S.A. Nazarov, On the spectrum of the Steklov problem in peak-shaped domains. Trudy St.-Petersburg Mat. Obshch. 14 (2008) 103-168. (English transl. Am. Math. Soc. Transl Ser. 2.) | MR 2584396 | Zbl 1183.35212

[32] S.A. Nazarov. On the essential spectrum of boundary value problems for systems of differential equations in a bounded peak-shaped domain. Funkt. Anal. i Prilozhen. 43 (2009) 55-67. (English transl. Funct. Anal. Appl. 43 (2009).) | MR 2503865 | Zbl 1271.35055

[33] S.A. Nazarov and K. Pileckas, On steady Stokes and Navier-Stokes problems with zero velocity at infinity in a three-dimensional exterior domain. Journal of Mathematics of Kyoto University 40 (2000) 475-49. | MR 1794517 | Zbl 0976.35051

[34] S.A. Nazarov and B.A. Plamenevskii, Radiation principles for self-adjoint elliptic problems. Probl. Mat. Fiz. 13. 192-244. Leningrad: Leningrad Univ. 1991 (Russian). | MR 1341639

[35] S.A. Nazarov and B.A. Plamenevskii, Elliptic problems in domains with piecewise smooth boundaries. Walter be Gruyter, Berlin, New York (1994). | MR 1283387 | Zbl 0806.35001

[36] S.A. Nazarov and O.R. Polyakova, Asymptotic behavior of the stress-strain state near a spatial singularity of the boundary of the beak tip type. Prikl. Mat. Mekh. 57 (1993) 130-149. (English transl. J. Appl. Math. Mech. 57 (1993) 887-902.) | MR 1262071 | Zbl 0794.73011

[37] S.A. Nazarov and S.A. Taskinen, On the spectrum of the Steklov problem in a domain with a peak. Vestnik St. Petersburg Univ. Math. 41 (2008) 45-52. | MR 2406898 | Zbl 1171.35086

[38] S.A. Nazarov and J. Taskinen, On essential and continuous spectra of the linearized water-wave problem in a finite pond. Math. Scand. 106 (2009) 1-20. | MR 2603466 | Zbl 1191.35016

[39] J. Peetre, Another approach to elliptic boundary problems. Comm. Pure. Appl. Math. 14 (1961) 711-731. | MR 171069 | Zbl 0104.07303

[40] B.A. Plamenevskii, The asymptotic behavior of the solutions of quasielliptic differential equations with operator coefficients. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 1332-1375. | MR 372656 | Zbl 0285.35029

[41] J.J. Stoker, Water waves. The Mathematical Theory with Applications. Reprint of the 1957 original. John Wiley, New York (1992). | MR 1153414 | Zbl 0812.76002

[42] F. Ursell, Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47 (1951) 347-358. | MR 41604 | Zbl 0043.40704