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Global regularity of wave maps. II: Small energy in two dimensions. (English) Zbl 1020.35046

Summary: 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 are smooth and has small norm in the critical Sobolev space \(H^{n/2}\), in all dimensions \(n\geq 2\). This generalizes the results in the prequel [Int. Math. Res. Not. 2001, No. 6, 299-328 (2001; Zbl 0983.35080)] 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.

MSC:

35L55 Higher-order hyperbolic systems
58E20 Harmonic maps, etc.
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 0983.35080
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