Abstract
We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ∂Ω of the domain as the square root of the distance to ∂Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.
Similar content being viewed by others
References
S. Yu. Dobrokhotov and V. E. Nazaikinskii, “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation,” Mat. Zametki 100 (5), 710–731 (2016) [Math. Notes 100 (5–6), 695–713 (2016)].
V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1988; Marcel Dekker, New York, 1971).
M. S. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space (Izd. Leningrad. Univ., Leningrad, 1980; Kluwer Academic Publishers, Dordrecht, 1987).
S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi, “Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity. I,” Russ. J. Math. Phys. 17 (4), 434–447 (2010).
S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi, “Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data,” Algebra Anal. 22 (6), 67–90 (2010) [St. Petersburg Math. J. 22 (6), 895–911 (2011)].
S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi, “Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data,” Russ. J. Math. Phys 20 (4), 389–401 (2013).
J. J. Stoker, WaterWaves. TheMathematical Theory with Applications (JohnWiley and Sons, New York, 1958).
E. N. Pelinovskii, Hydrodynamics of Tsunami Waves (Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, 1996) [in Russian].
E. N. Pelinovsky and R. Kh. Mazova, “Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles,” Natural Hazards 6 (3), 227–249 (1992).
S. Yu. Dobrokhotov, D. S. Minenkov, V. E. Nazaikinskii, and B. Tirozzi, “Simple exact and asymptotic solutions of the 1D run-up problem over a slowly varying (quasiplanar) bottom,” in Theory and Applications in Mathematical Physics (World Sci., Singapore, 2015), pp. 29–47.
V. E. Nazaikinskii, “TheMaslov canonical operator on Lagrangian manifolds in the phase space corresponding to a wave equation degenerating on the boundary,” Mat. Zametki 96 (2), 261–276 (2014) [Math. Notes 96 (1–2), 248–260 (2014)].
V. P. Maslov, Théorie des perturbations et méthodes asymptotiques (Izd. Moskov. Univ., Moscow, 1965; Dunod, Paris, 1972).
V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics (Nauka, Moscow, 1976; Reidel, Dordrecht, 1981).
S. Yu. Dobrokhotov, G. N. Makrakis, V. E. Nazaikinskii, and T. Ya. Tudorovskii, “New formulas for Maslov’s canonical operator in a neighborhood of focal points and caustics in two-dimensional semiclassical asymptotics,” Teoret. Mat. Fiz. 177 (3), 355–386 (2013) [Theoret. and Math. Phys. 177 (3), 1579–1605 (2013)].
V. E. Nazaikinskii, “On the representation of localized functions in R2 by the Maslov canonical operator,” Mat. Zametki 96 (1–2), 88–100 (2014) [Math. Notes 96 (1–2), 99–109 (2014)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text© S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. A. Tolchennikov, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 5, pp. 700–715.
Rights and permissions
About this article
Cite this article
Dobrokhotov, S.Y., Nazaikinskii, V.E. & Tolchennikov, A.A. Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach. Math Notes 101, 802–814 (2017). https://doi.org/10.1134/S0001434617050066
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434617050066