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Stability of Fractionally Dissipative 2D Quasi-geostrophic Equation with Infinite Delay

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Abstract

In this paper, fractionally dissipative 2D quasi-geostrophic equations with an external force containing infinite delay is considered in the space \(H^s\) with \(s\ge 2-2\alpha \) and \(\alpha \in (\frac{1}{2},1)\). First, we investigate the existence and regularity of solutions by Galerkin approximation and the energy method. The continuity of solutions with respect to initial data and the uniqueness of solutions are also established. Then we prove the existence and uniqueness of a stationary solution by the Lax–Milgram theorem and the Schauder fixed point theorem. Using the classical Lyapunov method, the construction method of Lyapunov functionals and the Razumikhin–Lyapunov technique, we analyze the local stability of stationary solutions. Finally, the polynomial stability of stationary solutions is verified in a particular case of unbounded variable delay.

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Acknowledgements

The authors express their sincere thanks to the anonymous reviewers for their careful reading of the paper, giving valuable comments and suggestions. It is their contributions that greatly improve the paper. The authors also thank the editors for their kind help.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, China

    Tongtong Liang & Yejuan Wang

  2. Depto. de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012, Seville, Spain

    Tomás Caraballo

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  1. Tongtong Liang
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  2. Yejuan Wang
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  3. Tomás Caraballo
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Correspondence to Yejuan Wang.

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This work was supported by NSF of China (Grant No. 41875084), and by FEDER and Ministerio de Ciencia, Innovación y Universidades of Spain (Grant PGC2018-096540-B-I00) and Junta de Andalucia (Grant US-1254251).

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Liang, T., Wang, Y. & Caraballo, T. Stability of Fractionally Dissipative 2D Quasi-geostrophic Equation with Infinite Delay. J Dyn Diff Equat 33, 2047–2074 (2021). https://doi.org/10.1007/s10884-020-09883-y

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  • DOI: https://doi.org/10.1007/s10884-020-09883-y

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