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Stability Analysis of Stochastic 3D Lagrangian-Averaged Navier-Stokes Equations with Infinite Delay

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Abstract

The asymptotic behaviour of stochastic three-dimensional Lagrangian-averaged Navier-Stokes equations with infinite delay and nonlinear hereditary noise is analysed. First, using Galerkin’s approximations and the monotonicity method, we prove the existence and uniqueness of solutions when the non-delayed external force is locally integrable and the delay terms are globally Lipschitz continuous with an additional assumption. Next, we show the existence and uniqueness of stationary solutions to the corresponding deterministic equation via the Lax-Milgram and the Schauder theorems. Later, we focus on the stability properties of stationary solutions. To begin with, we discuss the local stability of stationary solutions for general delay terms by using a direct method and then apply the abstract results to two kinds of infinite delays. Besides, the exponential stability of stationary solutions is also established in the case of unbounded distributed delay. Moreover, we investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing appropriate Lyapunov functionals. Eventually, we establish criteria on the polynomial asymptotic stability of stationary solutions for the special case of proportional delay.

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Acknowledgements

The research of the first author has been partially supported by CSC (China Scholarship Council, No. 202006990054) for 1 year’s study abroad at the University of Sevilla. The research of the first three authors has been partially supported by the National Natural Science Foundation of China (No. 11571283). The research of the first and fourth authors has been partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and by Junta de Andalucía (Consejería de Economía y Conocimiento) under project US-1254251.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Shuang Yang, Yangrong Li & Qiangheng Zhang

  2. Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/Tarfia s/n, 41012, Sevilla, Spain

    Shuang Yang & Tomás Caraballo

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  1. Shuang Yang
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  4. Tomás Caraballo
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Correspondence to Tomás Caraballo.

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Yang, S., Li, Y., Zhang, Q. et al. Stability Analysis of Stochastic 3D Lagrangian-Averaged Navier-Stokes Equations with Infinite Delay. J Dyn Diff Equat 35, 3011–3054 (2023). https://doi.org/10.1007/s10884-022-10244-0

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