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Global regularity of wave maps. I: Small critical Sobolev norm in high dimension. (English) Zbl 0983.35080

This paper considers wave maps from a Minkowski space \({\mathbb R}^{n+1}\) to a Euclidean sphere \(S^{m-1}\). Regularity is measured by the \(H^s\) norm modulo constants (the author writes \(H^s\) for \(H^s/{\mathbb R}\) for convenience). The main result states that if \(n\geq 5\), \(s>n/2\), and if the Cauchy data belong to \(H^s\times H^{s-1}\) and are small in \(\dot{H}^{n/2}\times \dot{H}^{n/2-1}\), then the solution to the initial value problem remains in \(H^s\times H^{s-1}\) for all time. The idea of the proof rests on the following: (i) it suffices to handle small perturbations of smooth maps; (ii) the linearization may be simplified by a change of frame in the target, which can be computed recursively using the Littlewood-Paley decomposition.

MSC:

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