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Bejenaru, I.; Ionescu, A. D.; Kenig, C. E.; Tataru, D.
Annals of mathematics, 05/2011, Letnik: 173, Številka: 3Journal Article
We consider the Schrödinger map initial-value problem $\cases \partial _{t}\phi =\phi \times \Delta \phi \,\text{on}\,{\Bbb R}^{d}\times {\Bbb R}, & \\ \phi (0)=\phi _{0}, &\endcases $ where ϕ: ℝ d × ℝ → 𝕊 2 ↪ ℝ 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 ∈ C(ℝ: $H_{Q}^{\infty}$ ), Q ∈ 𝕊 2 , provided that the data ϕ0 ∈ $H_{Q}^{\infty}$ is smooth and satisfies the smallness condition ∥ϕ0−Q∥ Ḣd/2 ⪡ 1. We prove also that the solution operator extends continuously to the space of data in Ḣ d/2 ∩ $\dot{H}_{Q}^{d/2-1}$ with small Ḣ d/2 norm.
