Global Schrödinger maps in dimensions d ≥ 2: Small data in the critical sobolev spaces

I. Bejenaru, A. D. Ionescu, C. E. Kenig, D. Tataru

Research output: Contribution to journalArticlepeer-review

66 Scopus citations

Abstract

We consider the Schrödinger map initial-value problem, where φR{double-struck}d × R{double-struck} → S{double-struck}2 → R{double-struck}3 is a smooth function. In all dimensions d ≥ 2, we prove that the Schrödinger map initial-value problem admits a unique global smooth solution φ ε C(R{double-struck} : HQ), Q ε S{double-struck}2, provided that the data φo ε HQ is smooth and satisfies the smallness condition ||φ0 - Q||Hd/2 << 1. We prove also that the solution operator extends continuously to the space of data in H.d/2 ∩ HQ d/2-1 with small H d/2. norm.

Original languageEnglish (US)
Pages (from-to)1443-1506
Number of pages64
JournalAnnals of Mathematics
Volume173
Issue number3
DOIs
StatePublished - May 2011

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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