Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-20T10:22:33.472Z Has data issue: false hasContentIssue false
  • English
  • Français

Radiation fields and hyperbolic scattering theory

Published online by Cambridge University Press:  24 October 2008

F. G. Friedlander
F. G. Friedlander
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

Let u(x, t), where x ∈ 3, t ∈ , be a C∞ solution of the wave equation on {|x| > a} × for some a > 0, and suppose that u = 0 for |x|> a, t < 0. One then also has u = 0 for |x| > a + t, t > 0, so that u(.,t) can be thought of as a wave expanding into a previously undisturbed medium. One can now ask for a description of the asymptotic behaviour of u as |x| → ∞. It turns out that there is a v0(θ, τ) ∈ C∞ (S2 × ) such that

in the topology of C∞ (S2 × ). This limit may be called the radiation field of the expanding wave u. (See (2) and the earlier papers quoted there.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 3 , November 1980 , pp. 483 - 515
Copyright
Copyright © Cambridge Philosophical Society 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

(1)Friedlander, F. G.On the radiation fields of pulse solutions of the wave equation: III. Proc. Roy. Soc. London, Ser. A 299 (1967), 386–394.Google Scholar
(2)Friedlander, F. G.An inverse problem for radiation fields. Proc. London Math. Soc. (3) 27 (1973), 551–576.CrossRefGoogle Scholar
(3)Friedlander, F. G.The wave equation on a curved space-time (Cambridge University Press, 1975).Google Scholar
(4)Hörmander, L. Uniqueness theorems and estimates for normally hyperbolic partial differential equations of the second order. Comptes Rendus, 12ième Congr. Mathem. Scand., Lund, 1953.Google Scholar
(5)Iwasaki, N.Local decay of solutions for symmetric hyperbolic systems with dissipative and coercive boundary conditions in exterior domains. Publ. RIMS, Kyoto Univ. 5 (1969), 193–218.CrossRefGoogle Scholar
(6)Lax, P. D. and Phillips, R. S.Scattering Theory (New York and London, Academic Press, 1967).Google Scholar
(7)Lax, P. D. and Phillips, R. S.Scattering theory for the acoustic equation in an even number of space dimensions. Indiana Univ. Math. J. 22 (1972), 101–134.CrossRefGoogle Scholar
(8)Lax, P. D. and Phillips, R. S. Scattering theory for automorphic functions. Ann. of Math. Studies 87 (Princeton, 1976).Google Scholar
(9)Penrose, R.Conformal treatment of infinity, in relativity, groups and topology, ed. De Witt, C. M. and De Witt, B. S. (New York, Gordon and Breach, 1964).Google Scholar
(10)Rauch, J. and Taylor, M.Penetrations into shadow regions and unique continuation properties in hyperbolic mixed problems. Indiana Univ. Math. J. 22 (1972), 277–285.CrossRefGoogle Scholar