Dynamics of Partial Differential Equations

Volume 15 (2018)

Number 4

Asymptotic stability of harmonic maps between 2D hyperbolic spaces under the wave map equation. II. Small energy case

Pages: 283 – 336

DOI: http://dx.doi.org/10.4310/DPDE.2018.v15.n4.a3

Authors

Ze Li (Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China)

Xiao Ma (Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China)

Lifeng Zhao (Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China)

Abstract

In this paper, we prove that the small energy harmonic maps from $\mathbb{H}^2$ to $\mathbb{H}^2$ are asymptotically stable under the wave map equation in the subcritical perturbation class. This result may be seen as an example supporting the soliton resolution conjecture for geometric wave equations without equivariant assumptions on the initial data. In this paper, we construct Tao’s caloric gauge in the case when nontrivial harmonic map occurs. With the “dynamic separation” the master equation of the heat tension field appears as a semilinear magnetic wave equation. By the endpoint and weighted Strichartz estimates for magnetic wave equations obtained by the first author [38], the asymptotic stability follows by a bootstrap argument.

Keywords

wave map equation, hyperbolic spaces, asymptotic stability, harmonic maps, curved spacetime

2010 Mathematics Subject Classification

35L70, 35Q75, 51P05

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Received 17 August 2017