Equivariant Schrödinger maps in two spatial dimensions: The $\mathbb{H}^{2}$ target

EQUIVARIANT SCHR ODINGER MAPS IN TWO SPATIAL DIMENSIONS: THE H 2 TARGET arXiv:1212.2566v1 [math.AP] 11 Dec 2012 I. BEJENARU, A. IONESCU, C. KENIG, AND D. TATARU Abstract. We consider equivariant solutions for the Schr¨odinger map problem from R 2+1 to H 2 with finite energy and show that they are global in time and scatter. 1. Introduction The Schr¨odinger map equation in R 2+1 with values into S µ ⊂ R 3 is given by u t = u × µ ∆u, u(0) = u 0 where µ = ±1, the connected Riemannian manifolds S µ , S 1 = S 2 = {y = (y 0 , y 1 , y 2 ) ∈ R 3 : y 1 2 + y 2 2 + y 3 2 = 1}; S −1 = H 2 = {y = (y 0 , y 1 , y 2 ) ∈ R 3 : −y 1 2 − y 2 2 + y 3 2 = 1, y 3 > 0}, with the Riemannian structures induced by the Euclidean metric g 1 = dy 0 2 + dy 1 2 + dy 2 2 on S 1 , respectively the Minkowski metric g −1 = −dy 0 2 + dy 1 2 + dy 2 2 on S −1 . Thus S 1 is the 2-dimensional sphere S 2 , while S −1 is the 2-dimensional hyperbolic space H 2 . With η µ = diag(1, 1, µ), the cross product × µ is defined by v × µ w := η µ · (v × w). This equation admits a conserved energy, Z E(u) = |∇u| 2 µ dx 2 R 2 and is invariant with respect to the dimensionless scaling u(t, x) → u(λ 2 t, λx). The energy is invariant with respect to the above scaling, therefore the Schr¨odinger map equation in R 2+1 is energy critical. The local theory for classical data was established in [25] and [21]. We recall Theorem 1.1 (McGahagan). If u 0 ∈ H ˙ 1 ∩ H ˙ 3 then there exists a time T > 0, such that (1.1) has a unique solution in L ∞ t ([0, T ] : H ∩ H ). The local and global in time of the Schr¨odinger map problem with small data has been intensely studied for the case µ = 1 corresponding to S 2 as target, see [3], [4], [5], [6], [9], [15], [16]. The state of the art result for the problem with small data was established by the authors in [6] where they proved that classical solutions (and in fact rough solutions too) with small energy are global in time. These results are expected to extend to the case µ = −1, corresponding to H 2 as a target. I.B. was supported in part by NSF grant DMS-1001676. A. I. was partially supported by a Packard Fellowship and NSF grant DMS-1065710. C.K. was supported in part by NSF grant DMS-0968742. D.T. was supported in part by the Miller Foundation and by NSF grant DMS-0801261.