Geometric renormalization of large energy wave maps

Tao, Terence

Journées équations aux dérivées partielles, (2004), p. 1-32 / Harvested from Numdam

Résumé

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.

Détail

Publié le : 2004-01-01

MR 2135366 | Zbl 02161537

DOI : https://doi.org/10.5802/jedp.11
Classification:  35J10

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