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

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 ∩ HQd/2-1 with small H d/2. norm.