Global Regularity of Wave Maps¶II. Small Energy in Two Dimensions

Abstract:

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 , in all dimensions n≥ 5. This generalizes the results in the prequel [40] of this paper, which addressed the high-dimensional case n≥ 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.