Global attractor of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics

Abstract

The objective of this paper is to study the existence of a global attractor in $$(H^2(\Omega ))^3\cap V$$ for the three-dimensional autonomous primitive equations of large-scale ocean and atmosphere dynamics. According to the regularity results for the Stokes-type system in cylinder-type domains established by Ziane (Appl Anal 58(3–4):263–292, 1995), we can only obtain the existence of an absorbing set in $$(H^2(\Omega ))^3\cap V$$ such that the compactness of the semigroup in $$(H^2(\Omega ))^3\cap V$$ cannot be proved by the Sobolev compactness embedding theorem. Therefore, in order to obtain the existence of a global attractor in $$(H^2(\Omega ))^3\cap V,$$ we carry out some a priori estimates of strong solutions to establish the asymptotical compactness of the semigroup in $$(H^2(\Omega ))^3\cap V$$ by asymptotic a priori estimate.