Dynamics of the three-dimensional primitive equations of large-scale atmosphere

Abstract

The main objective of this paper is to study the long-time behavior of weak solutions for the three-dimensional primitive equations of large-scale ocean. Due to the low regularity of the triple nonlinearity term, it is very difficult to obtain the uniqueness of weak solutions such that we can not study the long-time behavior of weak solutions by using the classical theory of infinite-dimensional dynamical systems. Inspired by the idea of ℓ-trajectory in [1], we introduce a new phase space $X_{\ell }$ (see Section 3.1 for the definition of $X_{\ell }$) on which the translation semigroup $\{L(t){\}}_{t\ge 0}$ can be imposed naturally. We first establish the finite-dimensional global attractor in $X_{\ell }$ by using the classical theory of infinite-dimensional dynamical systems, via a Lipschitz continuous mapping from $X_{\ell }$ into H(see Section 2 for the definition of H), we obtain the corresponding finite-dimensional global attractor in H.

Keywords:

• Primitive equation
• global attractor
• the method of ℓ-trajectories
• fractal dimension
• Aubin–Lions compactness lemma

2010 Mathematics Subject Classifications:

• 35Q35
• 35B40
• 37C60

Acknowledgments

This work was supported by the National Science Foundation of China Grant (11401459,11871389), the Natural Science Foundation of Shaanxi Province (2018JM1012) and the Fundamental Research Funds for the Central Universities (xjj2018088).

Disclosure statement

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

Additional information

Funding

This work was supported by the National Science Foundation of China [grant numbers 11401459, 11871389], the Natural Science Foundation of Shaanxi Province [grant number 2018JM1012] and the Fundamental Research Funds for the Central Universities [grant number xjj2018088].