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Linear Inviscid Damping for the \(\beta \)-Plane Equation

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Abstract

In this paper, we study the linear inviscid damping for the linearized \(\beta \)-plane equation around shear flows. We develop a new method to give the explicit decay rate of the velocity for a class of monotone shear flows. This method is based on the space-time estimate and the vector field method in the sprit of the wave equation. For general shear flows including the Sinus flow, we also prove the linear damping by establishing the limiting absorption principle, which is based on the compactness method introduced by Wei et al. (Ann PDE 5:3, 2019). The main difficulty is that the Rayleigh–Kuo equation has more singular points due to the Coriolis effects so that the compactness argument becomes more involved and delicate.

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Acknowledgements

H. Zhu would like to thank School of Mathematical Science at Peking University, where part of this work was done when he was a visitor. Z. Zhang is partially supported by NSF of China under Grant 11425103.

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Authors and Affiliations

  1. School of Mathematical Science, Peking University, 100871, Beijing, People’s Republic of China

    Dongyi Wei & Zhifei Zhang

  2. Chern Institute of Mathematics, Nankai University, 300071, Tianjin, People’s Republic of China

    Hao Zhu

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  1. Dongyi Wei
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  2. Zhifei Zhang
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  3. Hao Zhu
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Correspondence to Zhifei Zhang.

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Communicated by C. Mouhot

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Wei, D., Zhang, Z. & Zhu, H. Linear Inviscid Damping for the \(\beta \)-Plane Equation. Commun. Math. Phys. 375, 127–174 (2020). https://doi.org/10.1007/s00220-020-03727-y

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