Equivariant Schrödinger map flow on two dimensional hyperbolic space

Jiaxi Huang; Youde Wang; Lifeng Zhao

doi:10.3934/dcds.2020184

Article Contents

2020, Volume 40, Issue 7: 4379-4425. Doi: 10.3934/dcds.2020184

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Equivariant Schrödinger map flow on two dimensional hyperbolic space

1.
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China
2.
College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
3.
Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, AMSS, and School of Mathematical Sciences, UCAS, Beijing 100190, China

^* Corresponding author: Lifeng Zhao
^* Corresponding author: Lifeng Zhao

Received: July 31, 2019

Revised: November 30, 2019

Published: April 2020

J. Huang and L. Zhao are partially supported by the NSFC Grant No. 11771415. Y. Wang is partially supported by the NSFC GRant No. 11731001

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Abstract

In this article, we consider the Schrödinger flow of maps from two dimensional hyperbolic space $ {{\mathbb{H}}}^2 $ to sphere $ {{\mathbb{S}}}^2 $. First, we prove the local existence and uniqueness of Schrödinger flow for initial data $ u_0\in\mathbf{H}^3 $ using an approximation scheme and parallel transport introduced by McGahagan [32]. Second, using the Coulomb gauge, we reduce the study of the equivariant Schrödinger flow to that of a system of coupled Schrödinger equations with potentials. Then we prove the global existence of equivariant Schrödinger flow for small initial data $ u_0\in\mathbf{H}^1 $ by Strichartz estimates and perturbation method.

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Mathematics Subject Classification: Primary: 35B65; Secondary: 35Q41.

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