Low regularity solutions for the wave map equation into the 2-D sphere
type: Item
instance of: scholarly article
title: Low regularity solutions for the wave map equation into the 2-D sphere
zbMATH Open document ID: 1063.35103
author: Vladimir Georgiev
author: Piero D'ancona
published in: Mathematische Zeitschrift
publication date: 2005-02-01T00:00:00Z
review text: The authors investigate the well posedeness of the Cauchy problem for the wave map equation \(\partial^{2}_{t}u- \Delta_{x}u+(| \partial_{t}u|^{2} - |\nabla_{x}u|^{2})u=F\) subject to the constraint \(| u |=1\) in two dimensions (\(u:{\mathbb{R}}\times{\mathbb{R}^{2}} \rightarrow {\mathbb{S}^{2}} \subset{\mathbb{R}^{3}}\)). They prove a nonuniqueness result and that in general the solutions may not depend continuously on \(F\). More precisely, using self-similar solutions, they construct two weak solutions of the homogeneous equation which coincide for \(t<0\) and which belong to \(C(\mathbb{R}, H^{1-\varepsilon}(\Omega))\) for all \(\varepsilon>0\) and all bounded \(\Omega\) and to \(L^{\infty}(\mathbb{R}, B^{1}_{2, \infty}(\Omega))\), \(B^{1}_{2, \infty}\) the Besov space, for all bounded \(\Omega\). They also construct functions \(F_{k}\) and weak solutions \(u_{k}\) which satisfy the same initial conditions such that the norm of \(F_{k}\) tends to \(0\) in \(L^{p}([0,T],L^{q}(\mathbb{R}^{2}))\), \(p\geq 1\), \(q\geq 2\), \({1\over{p}}+{2\over{q}}>2\), while the norm of \(u_{k}(T,x)-u(0,x)\) diverges to \(+\infty\) in \(H^{1}(\mathbb{R}^{2})\) for all \(T>0\).
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 58J45
zbMATH DE Number: 2131979
zbMATH Keywords: well posedness
zbMATH Keywords: self-similar solutions
zbMATH Keywords: ill-posedness
MaRDI profile type: Publication
full work available at URL: s00209-003-0558-3
OpenAlex ID: W2081483171
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DOI: 10.1007/S00209-003-0558-3
description: scientific article; zbMATH DE number 2131979