# Geometric renormalization of large energy wave maps

Terence Tao^{[1]}

- [1] Department of Mathematics, UCLA, Los Angeles CA 90095-1555

Journées Équations aux dérivées partielles (2004)

- page 1-32
- ISSN: 0752-0360

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topTao, Terence. "Geometric renormalization of large energy wave maps." Journées Équations aux dérivées partielles (2004): 1-32. <http://eudml.org/doc/10590>.

@article{Tao2004,

abstract = {There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the solution is large. In this report we present a different renormalization, based on the harmonic map heat flow, which works for large energy wave maps from two dimensions to hyperbolic spaces. We also observe an intriguing estimate of “non-concentration” type, which asserts roughly speaking that if the energy of a wave map concentrates at a point, then it becomes asymptotically self-similar.},

affiliation = {Department of Mathematics, UCLA, Los Angeles CA 90095-1555},

author = {Tao, Terence},

journal = {Journées Équations aux dérivées partielles},

keywords = {wave maps; harmonic maps; renormalization; Coulomb gauge; Minkowski space},

language = {eng},

month = {6},

pages = {1-32},

publisher = {Groupement de recherche 2434 du CNRS},

title = {Geometric renormalization of large energy wave maps},

url = {http://eudml.org/doc/10590},

year = {2004},

}

TY - JOUR

AU - Tao, Terence

TI - Geometric renormalization of large energy wave maps

JO - Journées Équations aux dérivées partielles

DA - 2004/6//

PB - Groupement de recherche 2434 du CNRS

SP - 1

EP - 32

AB - There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the solution is large. In this report we present a different renormalization, based on the harmonic map heat flow, which works for large energy wave maps from two dimensions to hyperbolic spaces. We also observe an intriguing estimate of “non-concentration” type, which asserts roughly speaking that if the energy of a wave map concentrates at a point, then it becomes asymptotically self-similar.

LA - eng

KW - wave maps; harmonic maps; renormalization; Coulomb gauge; Minkowski space

UR - http://eudml.org/doc/10590

ER -

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