The wave map problem. Small data critical regularity

Rodnianski, Igor

Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 965, pp. 365-384.

Cet exposé présente le problème de l’application d’onde en se concentrant sur le travail révolutionnaire de T. Tao qui a montré que l’application d’onde, un analogue lorentzien dynamique d’une application harmonique, de l’espace de Minkowski dans la sphère avec une donnée initiale et une petite norme de Sobolev critique existe globalement en temps et reste lisse. Quand la dimension de l’espace de Minkowski est (2+1), la norme critique coïncide avec l’énergie, la seule quantité clairement conservée dans cette théorie lagrangienne. Comme conséquence, dans cette dimension, le résultat est une étape importante dans la preuve de la régularité globale pour n’importe quelle énergie, qui a été conjecturée quand la variété d’arrivée est de courbure négative. Ce travail a fait avancer notre compréhension des équations critiques et a déjà servi de catalyseur pour de nouveaux résultats pour des variétés d’arrivée plus générales et d’autres équations (Maxwell-Klein-Gordon, Yang-Mills).

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).

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