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

Abstract

We show that wave maps from Minkowski space ${\mathbb{R}}^{1+n}$ 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 $n\ge 2$. This generalizes the results in the prequel [math.AP/0010068] of this paper, which addressed the high-dimensional case $n\ge 5$. 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.

109 pages, no figures, submitted to Comm. Math. Phys. A technical error (U and phi need to be measured in slightly different spaces for induction purposes) has been corrected, and some other small errors fixed

References in corpus (1)

• Global regularity of wave maps I. Small critical Sobolev norm in high dimension

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