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Dynamics and Stability Analysis for Stochastic 3D Lagrangian-Averaged Navier–Stokes Equations with Infinite Delay on Unbounded Domains

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Abstract

This paper is devoted to investigating mean dynamics and stability analysis for stochastic 3D Lagrangian-averaged Navier–Stokes (LANS) equations driven by infinite delay on unbounded domains. We first prove the existence of a unique solution to stochastic 3D LANS equations with infinite delay when the non-delayed external force is locally integrable, the delay term is globally Lipschitz continuous and the nonlinear diffusion term is locally Lipschitz continuous. This enables us to define a mean random dynamical system. Besides, we find that such a dynamical system possesses a unique weak pullback mean random attractor, which is a minimal, weakly compact and weakly pullback attracting set. Furthermore, we prove the existence and uniqueness of stationary solutions (equilibrium solutions) to the corresponding deterministic equation via the classical Galerkin method, the Lax–Milgram and the Brouwer fixed theorems. The stability properties of stationary solutions are also considered. By a direct approach, we first show the local stability of stationary solutions when the delay term has a general form and then apply the abstract results to two kinds of infinite delays. Second, we establish the exponential stability of stationary solutions in the case of unbounded distributed delay. Third, we investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing appropriate Lyapunov functionals. Eventually, we discuss the polynomial asymptotic stability in the particular case of proportional delay.

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Acknowledgements

This work was done when Shuang Yang visited the Department of Differential Equations and Numerical Analysis at the University of Sevilla. She would like to express her thanks to all people there for their kind hospitality.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Shuang Yang & Yangrong Li

  2. Departamento 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. Shuang Yang
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  2. Tomás Caraballo
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  3. Yangrong Li
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Correspondence to Yangrong Li.

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Shuang Yang was supported by the China Scholarship Council (CSC No. 202006990054). The research of Shuang Yang and Tomás Caraballo have 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 Projects US-1254251 and P18-FR-4509. The research of the first and third authors has been partially supported by the National Natural Science Foundation of China (No. 11571283).

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Yang, S., Caraballo, T. & Li, Y. Dynamics and Stability Analysis for Stochastic 3D Lagrangian-Averaged Navier–Stokes Equations with Infinite Delay on Unbounded Domains. Appl Math Optim 89, 11 (2024). https://doi.org/10.1007/s00245-023-10081-7

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