On the stability of stationary wave maps
type: Item
instance of: scholarly article
title: On the stability of stationary wave maps
zbMATH Open document ID: 0877.58054
author: J. Shatah
author: A. Shadi Tahvildar-Zadeh
published in: Communications in Mathematical Physics
publication date: 1997-12-14T00:00:00Z
review text: A wave map is a map \(U\) from a Lorentzian manifold \((M^{m+1},g)\) into a Riemannian manifold \((N^n,h)\) which is a critical point of the Lagrangian \[ {\mathcal L} (U)= {\textstyle {1\over 2}} \int_MTr_gU^*h ={\textstyle {1\over 2}} \int_M|DU|^2 ={\textstyle {1\over 2}} \int_M g^{\alpha\beta} h_{ab} D_\alpha U^a D_\beta U^b. \] In the article under review the case \(M=S^2 \times\mathbb{R}\) and \(N=S^2\) is considered. The existence of stationary maps generated by the symmetry groups and their stability under equivariant perturbations are proved. As consequence of this stability result it is obtained the existence of a large set of initial data, with no degree or energy restrictions, for which the Cauchy problem is globally well posed.
Mathematics Subject Classification ID: 58J45
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1019456
zbMATH Keywords: equivariant wave maps
zbMATH Keywords: equivariant perturbations
zbMATH Keywords: stability
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: s002200050089
OpenAlex ID: W2003447438
DOI: 10.1007/S002200050089
description: scientific article; zbMATH DE number 1019456