The wave map problem. Small data critical regularity

Igor Rodnianski

Séminaire Bourbaki (2005-2006)

  • Volume: 48, page 365-384
  • ISSN: 0303-1179

Abstract

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The paper provides a description of the wave map problem with a specific focus on the breakthrough work of T. Tao which showed that a wave map, a dynamic lorentzian analog of a harmonic map, from Minkowski space into a sphere with smooth initial data and a small critical Sobolev norm exists globally in time and remains smooth. When the dimension of the base Minkowski space is ( 2 + 1 ) , the critical norm coincides with energy, the only manifestly conserved quantity in this (lagrangian) theory. As a consequence, in this dimension the result is an important step in establishing global regularity at all energies, conjectured when the target manifold is negatively curved. The work advanced our understanding of the critical equations and already has been a catalyst for the new results for general target manifolds and other equations (Maxwell-Klein-Gordon, Yang-Mills).

How to cite

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Rodnianski, Igor. "The wave map problem. Small data critical regularity." Séminaire Bourbaki 48 (2005-2006): 365-384. <http://eudml.org/doc/252156>.

@article{Rodnianski2005-2006,
abstract = {The paper provides a description of the wave map problem with a specific focus on the breakthrough work of T. Tao which showed that a wave map, a dynamic lorentzian analog of a harmonic map, from Minkowski space into a sphere with smooth initial data and a small critical Sobolev norm exists globally in time and remains smooth. When the dimension of the base Minkowski space is $(2+1)$, the critical norm coincides with energy, the only manifestly conserved quantity in this (lagrangian) theory. As a consequence, in this dimension the result is an important step in establishing global regularity at all energies, conjectured when the target manifold is negatively curved. The work advanced our understanding of the critical equations and already has been a catalyst for the new results for general target manifolds and other equations (Maxwell-Klein-Gordon, Yang-Mills).},
author = {Rodnianski, Igor},
journal = {Séminaire Bourbaki},
keywords = {wave map; critical regularity; renormalization},
language = {eng},
pages = {365-384},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {The wave map problem. Small data critical regularity},
url = {http://eudml.org/doc/252156},
volume = {48},
year = {2005-2006},
}

TY - JOUR
AU - Rodnianski, Igor
TI - The wave map problem. Small data critical regularity
JO - Séminaire Bourbaki
PY - 2005-2006
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 48
SP - 365
EP - 384
AB - The paper provides a description of the wave map problem with a specific focus on the breakthrough work of T. Tao which showed that a wave map, a dynamic lorentzian analog of a harmonic map, from Minkowski space into a sphere with smooth initial data and a small critical Sobolev norm exists globally in time and remains smooth. When the dimension of the base Minkowski space is $(2+1)$, the critical norm coincides with energy, the only manifestly conserved quantity in this (lagrangian) theory. As a consequence, in this dimension the result is an important step in establishing global regularity at all energies, conjectured when the target manifold is negatively curved. The work advanced our understanding of the critical equations and already has been a catalyst for the new results for general target manifolds and other equations (Maxwell-Klein-Gordon, Yang-Mills).
LA - eng
KW - wave map; critical regularity; renormalization
UR - http://eudml.org/doc/252156
ER -

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