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Stochastic 3D Globally Modified Navier–Stokes Equations: Weak Attractors, Invariant Measures and Large Deviations

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Abstract

This paper is mainly concerned with the asymptotic dynamics of non-autonomous stochastic 3D globally modified Navier–Stokes equations driven by nonlinear noise. Based on the well-posedness of such equations, we first show the existence and uniqueness of weak pullback mean random attractors. Then we investigate the existence of (periodic) invariant measures, the zero-noise limit of periodic invariant measures and their limit as the modification parameter \(N\rightarrow N_0\in (0,+\infty )\). Furthermore, under weaker conditions, we obtain the existence of invariant measures as well as their limiting behaviors when the external term is independent of time. Finally, by using weak convergence method, we establish the large deviation principle for the solution processes.

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Acknowledgements

The authors are grateful to the editor and referees for their very valuable suggestions and comments.

Funding

The work is partially supported by the NNSF of China (11471190, 11971260), the SDNSF (ZR2014AM002), the Spanish Ministerio de Ciencia e Innovación under project PID2021-122991NB-C21, and Junta de Andalucía (Spain) under Project P18-FR-4509.

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Authors and Affiliations

  1. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/Tarfia s/n, 41012, Sevilla, Spain

    Tomás Caraballo

  2. Department of Mathematics, Wenzhou University, Wenzhou, 325035, China

    Tomás Caraballo

  3. School of Mathematics, Shandong University, Jinan, 250100, China

    Zhang Chen & Dandan Yang

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  1. Tomás Caraballo
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  2. Zhang Chen
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Correspondence to Dandan Yang.

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Caraballo, T., Chen, Z. & Yang, D. Stochastic 3D Globally Modified Navier–Stokes Equations: Weak Attractors, Invariant Measures and Large Deviations. Appl Math Optim 88, 74 (2023). https://doi.org/10.1007/s00245-023-10050-0

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