Global regularity of wave maps II. Small energy in two dimensions

Details Global regularity of wave maps II. Small energy in two dimensions Global regularity of wave maps II. Small energy in two dimensions

2000-11-22

arxiv.org/abs/math/0011173v3

Mathematics

Analysis of PDEs

109 pages, no figures, submitted to Comm. Math. Phys. A technical error (U and phi need to be measured in slightly different s

Scientific paper

10.1007/PL00005588

We show that wave maps from Minkowski space $\R^{1+n}$ to a sphere $S^{m-1}$ are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$, in all dimensions $n \geq 2$. This generalizes the results in the prequel [math.AP/0010068] of this paper, which addressed the high-dimensional case $n \geq 5$. In particular, in two dimensions we have global regularity whenever the energy is small, and global regularity for large data is thus reduced to demonstrating non-concentration of energy.

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