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Hausdorff Sub-norm Spaces and Continuity of Random Attractors for Bi-stochastic g-Navier–Stokes Equations with Respect to Tempered Forces

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Abstract

We study the continuity of pullback random attractors \(A_\lambda \), where the parameter belongs to a complete metric space \(\Lambda \). By a Hausdorff sub-norm space, we mean the collection of all nonempty compact subsets of the state space, equipped by the Hausdorff metric as well as a sub-norm. Under weaker conditions, we prove that the binary map \((\lambda , s)\rightarrow A_\lambda (s,\theta _s\omega )\) is continuous at all points of \(\Lambda ^*\times {\mathbb {R}}\) with respect to the Hausdorff metric, where \(\Lambda ^*\) is residual and dense in \(\Lambda \), and that the binary map is continuous under the sub-norm if and only if each fibre of attractors is a point. The proofs of these results are based on the discussion of continuous operators on the Hausdorff sub-norm space as well as the theory of Baire category. For the g-Navier–Stokes equation driven by random density, stochastic noise and time-dependent forces, we establish the residual continuity and full upper semi-continuity of pullback random attractors on the Fréchet space formed from all backward tempered forces.

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Acknowledgements

Li was supported by National Natural Science Foundation of China Grant 11571283. The authors would like to thank the anonymous referees for their careful reading and helpful comments which lead to an improvement of the manuscript.

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

    Yangrong Li & Shuang Yang

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  1. Yangrong Li
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Li, Y., Yang, S. Hausdorff Sub-norm Spaces and Continuity of Random Attractors for Bi-stochastic g-Navier–Stokes Equations with Respect to Tempered Forces. J Dyn Diff Equat 35, 543–574 (2023). https://doi.org/10.1007/s10884-021-10026-0

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