On the asymptotic behavior of spherically symmetric wave maps
type: Item
instance of: scholarly article
title: On the asymptotic behavior of spherically symmetric wave maps
zbMATH Open document ID: 0791.58105
DOI: 10.1215/S0012-7094-93-07103-7
author: Demetrios Christodoulou
author: A. Shadi Tahvildar-Zadeh
published in: Duke Mathematical Journal
publication date: 1994-07-14T00:00:00Z
review text: ``Wave maps are maps from an Einsteinian manifold \(M\), i.e. a manifold with a metric \(g\) of index 1, into a Riemannian manifold \(N\), i.e. one with a positive definite metric \(h\). Formally, they are the critical points \(U: M\to N\) of the Lagrangian \[ {\mathcal L}[U] = {1\over 2} \int_ M h_{ab} \partial_ \mu U^ a \partial^ \mu U^ b. \] In a previous paper [the authors, Commun. Pure Appl. Math. 46, No. 7, 1041-1091 (1993; Zbl 0744.58071)] we proved the global regularity of the solution to the Cauchy problem for a spherically symmetric wave map in the case where \(M\) is the \((2+1)\)-dimensional Minkowski space-time and \(N\) satisfies the following two conditions. (C1) There exists an orthonormal frame of smooth vectorfields \(\Omega_ A\) on \(N\) whose structure functions \(e^ C_{AB}\) are bounded. (C2) For each \(p \in N\), let \(\Sigma(p,s)\) be the geodesic sphere of radius \(s\) centered at \(p\) and let \(k_{AB}\) be its second fundamental form. Then there exists constants \(c\) and \(C\) such that \[ s \lambda \geq c\quad\text{and}\quad s \Lambda \leq C(1+s), \] where \(\lambda\) and \(\Lambda\) are respectively the smallest and largest eigenvalues of \(k_{AB}\). The map \(U\) satisfies the Euler-Lagrange system that in local coordinates can be written as \[ \square U^ a + \Gamma^ a_{bc} \partial_ \mu U^ b \partial^ \mu U^ c = 0,\tag{1} \] where \(\square = -\partial^ 2_ t + \partial^ 2_ r + r^{-1} \partial_ r\), and \(\Gamma\)'s are the Christoffel symbols of the target \(N\). Here we obtain the asymptotic behavior of these solutions. In particular, it will be shown that the asymptotic behavior is like that of the linear wave equation.
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 58J45
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 495436
zbMATH Keywords: Einsteinian manifold
zbMATH Keywords: Lagrangian
zbMATH Keywords: wave map
zbMATH Keywords: Euler-Lagrange system
reviewed by: Gerald Warnecke
MaRDI profile type: Publication
cites work: On the regularity of spherically symmetric wave maps
full work available at URL: s0012-7094-93-07103-7
OpenAlex ID: W1987003062
description: scientific article; zbMATH DE number 495436