We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map under the Schrödinger maps evolution with respect to non-equivariant perturbations, provided obeys a suitable linearized stability condition. This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper [36]. © 2021 Wiley Periodicals LLC.

中文翻译：

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薛定谔映射演化下双曲平面调和映射的渐近稳定性

我们考虑域为双曲平面时薛定谔映射演化的柯西问题。与欧几里德平面上更广泛研究的情况相比，这个问题的一个有趣特征是存在丰富的新的有限能量调和映射族。这些是稳态解，因此在薛定谔映射的动力学中起着重要作用。本文的主要结果是在薛定谔映射演化下（部分）这样的调和映射的渐近稳定性。更准确地说，我们证明了关于非等变扰动的薛定谔映射演化下的有限能量等变调和映射的非线性渐近稳定性，前提是服从合适的线性化稳定性条件。已知此条件适用于所有在双曲平面中具有值的等变调和映射以及那些在球面中取值的映射的子集。该论文的主要技术成分之一是全局时间局部平滑和 Strichartz 估计，该估计是通过谐波映射周围线性化获得的算子，在配套论文 [36] 中得到证明。© 2021 Wiley Periodicals LLC。