Solutions of wave equations in the radiation regime

Authors: Piotr T. Chrusciel, O. Lengard
Comments: 70 pages, latex2e, several style files
Subj-class: Analysis of PDEs

We study the ``hyperboloidal Cauchy problem'' for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behaviour at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a $\lambda\phi^p$ nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal completions at null infinity, as well to a large class of equations with a similar non-linearity structure. We prove existence of solutions with controlled asymptotic behaviour, and asymptotic expansions for solutions when the initial data have such expansions. In particular we prove that polyhomogeneous initial data (satisfying compatibility conditions) lead to solutions which are polyhomogeneous at the conformal boundary Scri of the Minkowski space-time.

Download: ps, or pdf, or dvi

In this directory you can also find a dvi and ps file of the paper "Polyhomogeneous solutions of wave equations in the radiation regime" (the files proceedings.*), written in collaboration with my PhD student, O.Lengard. This has been published in the Proceedings of the conference Journées Equations aux dérivées partielles, Nantes , 5-9 june, 2000, N. Depauw, D. Robert, X. Saint Raymond, Eds., pp. III-1 - III-17 (2000).

A PDF file of the paper can be retrieved directly from the web site of the Proceedings

Further, the files runthese.* contain O.Lengard's thesis on those topics, as well as on related topics for Einstein's equations

Listing of Sat Feb 2 10:38:44 CET 2002:
Name (Last modified) Size
proceedings.dvi   (Aug 23 2000) 71420   (Aug 23 2000) 290988
runthese.dvi   (Nov 28 22:28) 653668
runthese.pdf   (Nov 28 22:29) 889282   (Nov 28 22:29) 1369428

[math/0506423] Polyhomogeneous solutions of nonlinear wave equations without corner conditions

Mathematics, abstract

From: Szymon {\L}\c{e}ski [view email]
Date: Tue, 21 Jun 2005 08:59:39 GMT   (72kb)

Polyhomogeneous solutions of nonlinear wave equations without corner conditions

Authors: Piotr T. Chrusciel, Szymon Leski
Subj-class: Analysis of PDEs
The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

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