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Global Existence and Scattering Behavior for One Dimensional Wave Maps into Riemannian Manifolds

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Abstract

This paper explores the global existence and scattering behavior for wave maps from Minkowski space \({\mathbb {R}}^{1+1}\) into general Riemannian manifolds, provided the initial data are small. In particular, we focus on the scattering fields of wave maps at the infinities, via which we conclude that the nonlinear scattering operator can be linearized as the corresponding linear scattering operator. This is accomplished by first exploiting the null-form structure in wave map equations, followed by a systematic analysis of weighted energy estimates.

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Acknowledgements

The first version of this paper was carried out during the author’s Ph.D. studies at Department of Mathematical Sciences and Yau Mathematical Sciences Center of Tsinghua University. The final stages of this paper were supported in part by the Natural Science Foundation of Jiangsu Province under Grant No. BK20220792; in part by the National Natural Science Foundation of China under Grant No. 12171267; and in part by the Science Climbing Program of Southeast University under Grant No. 4060692201/020.

Funding

The author was supported in part by the Natural Science Foundation of Jiangsu Province under Grant No. BK20220792; in part by the National Natural Science Foundation of China under Grant No. 12171267; and in part by the Science Climbing Program of Southeast University under Grant No. 4060692201/020.

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  1. School of Mathematics, Southeast University, No.2 SEU Road, Nanjing, 211189, People’s Republic of China

    Mengni Li

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Li, M. Global Existence and Scattering Behavior for One Dimensional Wave Maps into Riemannian Manifolds. Results Math 77, 164 (2022). https://doi.org/10.1007/s00025-022-01668-7

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