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

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

2000-10-07

arxiv.org/abs/math/0010068v3

Mathematics

Analysis of PDEs

24 pages, no figures, to appear, IMRN. The continuity argument has been once again simplified, some references added, and more

Scientific paper

We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$. A major difficulty, not present in the earlier results, is that the $\dot H^{n/2}$ norm barely fails to control $L^\infty$, potentially causing a logarithmic divergence in the nonlinearity; however, this can be overcome by using co-ordinate frames adapted to the wave map by approximate parallel transport. In the sequel of this paper we address the more interesting two-dimensional case, which is energy-critical.

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