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- Title
An alternative proof for small energy implies regularity for radially symmetric (1+2)-dimensional wave maps.
- Authors
Lai, Ning-An; Zhou, Yi
- Abstract
In this paper we are interested in showing an alternative and simple proof for small energy implies regularity to the Cauchy problem of radially symmetric wave maps from the (1 + 2) -dimensional Minkowski space to an arbitrary smooth Riemannian manifold M ⊂ R n with bounded first derivatives of the second fundamental form. Then, combining the classical works of Struwe (Math Z 242:407–414, 2002; Calc Var Partial Differ Equ 16:431–437, 2003), which gave a simple proof for non-concentration of energy to the corresponding problem, a new proof which we think should it be simple can be provided to the result of global regularity for radially symmetric (1 + 2) -dimensional wave maps, which was first obtained by Christodoulou and Tahvildar-Zadeh (Commun Pure Appl Math 46:1041–1091, 1993). Our method relies on basic energy estimates, based on a new div-curl type lemma developed by Zhou ((1 + 2)-dimensional radially symmetric wave maps revisit) and Wang-Zhou (Global well-posedness for radial extremal hypersurface equation in (1 + 3)-dimensional minkowski space-time in critical sobolev space; On proof of the Wei-Yue Ding's conjecture for Schrödinger map flow).
- Subjects
SOBOLEV spaces; MINKOWSKI space; CAUCHY problem; HARMONIC maps; RIEMANNIAN manifolds
- Publication
Calculus of Variations & Partial Differential Equations, 2024, Vol 63, Issue 2, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-023-02642-z