SCHRÖDINGER MAPS IN DIMENSIONS d ≥ 2 : SMALL DATA IN THE CRITICAL SOBOLEV SPACES
نویسنده
چکیده
We consider the Schrödinger map initial-value problem { ∂tφ = φ×∆φ on R × R; φ(0) = φ0, where φ : R × R → S →֒ R 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 ), Q ∈ S, provided that the data φ0 ∈ H∞ Q 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 ∩ Ḣ Q with small Ḣ norm.
منابع مشابه
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