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.
@incollection{JEDP_2004____A11_0, author = {Terence Tao}, title = {Geometric renormalization of large energy wave maps}, booktitle = {}, series = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {11}, pages = {1--32}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2004}, doi = {10.5802/jedp.11}, mrnumber = {2135366}, zbl = {02161537}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.11/} }
TY - JOUR AU - Terence Tao TI - Geometric renormalization of large energy wave maps JO - Journées équations aux dérivées partielles PY - 2004 SP - 1 EP - 32 PB - Groupement de recherche 2434 du CNRS UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.11/ DO - 10.5802/jedp.11 LA - en ID - JEDP_2004____A11_0 ER -
%0 Journal Article %A Terence Tao %T Geometric renormalization of large energy wave maps %J Journées équations aux dérivées partielles %D 2004 %P 1-32 %I Groupement de recherche 2434 du CNRS %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.11/ %R 10.5802/jedp.11 %G en %F JEDP_2004____A11_0
Terence Tao. Geometric renormalization of large energy wave maps. Journées équations aux dérivées partielles (2004), article no. 11, 32 p. doi : 10.5802/jedp.11. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.11/
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