Radiation conditions at the top of a rotational cusp in the theory of water-waves

Nazarov, Sergey A.; Taskinen, Jari

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 5, pp. 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.

[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 | Zbl

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

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

[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 | Zbl

[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

[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)

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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.) | EuDML | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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