Pullback exponential attractors for the three dimensional non-autonomous primitive equations of large scale ocean and atmosphere dynamics

The main objective of this paper is to study the existence of pullback exponential attractors for the three-dimensional non-autonomous primitive equations of large-scale ocean and atmosphere dynamics. Due to the shortage of the proof of the uniqueness of weak solutions, it is very difficult to define a solution process such that we cannot obtain the existence of pullback exponential attractors by the standard theory of pullback exponential attractor established in$\href{https://doi.org/10.3934/cpaa.2013.12.3047}{[\textrm{A.N. Carvalho and S. Sonner, } \textit{Commun. Pure Appl. Anal., } \textrm{12(6):3047–3071, 2013]}},$$\href{https://doi.org/10.1016/j.jmaa.2011.03.053}{[\textrm{R. Czaja and M. Efendiev, } \textit{J. Math. Anal. Appl. , } \textrm{381(2):748–765, 2011]}},$$\href{https://doi.org/10.1017/S030821050000408X}{[\textrm{M. Efendiev, S. Zelik, and A. Miranville, } \textit{ Proc. Roy. Soc. Edinb., } \textrm{Sect. A, 135(4):703–730, 2005]}},$$\href{https://doi.org/10.3934/DCDS.2010.26.1329}{[\textrm{J.A. Langa, A. Miranville, and J. Real, } \textit{Discrete Contin. Dyn. Syst., } \textrm{26(4):1329–1357, 2010]}}.$

Inspired by the idea of the method of $\ell$-trajectories, we will prove the existence of pullback exponential attractors by the abstract results established in$\href{https://doi.org/10.1002/mma.7413}{\textrm{ B. You, } \textit{Math. Meth. Appl. Sci., } \textrm{44(13):10361–10386, 2021]}}$.

Keywords

pullback exponential attractors, primitive equations, Aubin–Lions compactness lemma, the method of $\ell$-trajectories, trajectory space

2010 Mathematics Subject Classification

35B41, 35Q86, 37C60, 37L25, 37N10

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This work was supported by the National Science Foundation of China Grant (11871389, 11401459), and by the Fundamental Research Funds for the Central Universities (xzy012022008).

Received 19 September 2021

Accepted 26 October 2022

Published 30 August 2023