Existence and upper semicontinuity of random attractors for the 2D stochastic convective Brinkman–Forchheimer equations in bounded domains

Abstract

In this work, we discuss the large time behaviour of the solutions of two-dimensional stochastic convective Brinkman–Forchheimer (SCBF) equations on bounded domains. Under the functional setting $\mathbb{V}\hookrightarrow \mathbb{H}\hookrightarrow {\mathbb{V}}^{\mathrm{\prime }}$, where $\mathbb{H}$ and $\mathbb{V}$ are appropriate separable Hilbert spaces and the fact that the embedding $\mathbb{V}\hookrightarrow \mathbb{H}$ is compact, we establish the existence of random attractors in $\mathbb{H}$ for the stochastic flow generated by 2D SCBF equations perturbed by additive noise. We prove the upper semicontinuity of the random attractors for 2D SCBF equations in $\mathbb{H}$, when the coefficient of random term approaches zero. Moreover, we obtain the existence of random attractors in a more regular space $\mathbb{V}$, using the pullback flattening property. The existence of random attractors ensures the existence of invariant compact random set and hence we show the existence of an invariant measure for 2D SCBF equations. Finally, we also comment on the uniqueness of invariant measures.

Keywords:

• Stochastic convective Brinkman–Forchheimer equations
• cylindrical wiener process
• random attractors
• flattening property
• upper semicontinuity

Mathematics Subject Classification (2020):

• Primary 35B41
• 35Q35
• Secondary 37L55
• 37N10
• 35R60

Acknowledgements

The authors sincerely would like to thank the reviewer for his/her valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author would like to thank the Council of Scientific & Industrial Research (CSIR), India, for financial assistance (File No. 09/143(0938)/2019-EMR-I). M. T. Mohan would like to thank the Department of Science and Technology (DST), Govt of India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110).