The global attractor for the 3-D viscous primitive equations of large-scale moist atmosphere

Abstract

Absorbing ball in $H^{1}(\mho)$ is obtained for the strong solution to the three dimensional viscous moist primitive equations under the natural assumption $Q_{1},Q_{2}\in L^{2}(\mho)$ which is weaker than the assumption $Q_{1},Q_{2}\in H^{1}(\mho)$ in $\cite{GH2}$. In view of the structure of the manifold and the special geometry involved with vertical velocity, the continuity of the strong solution in $H^{1}(\mho)$ is established with respect to time and initial data. To obtain the existence of the global attractor for the moist primitive equations, the common method is to obtain the absorbing ball in $H^{2}(\mho)$ for the strong solution to the equations. But it is difficult due to the complex structure of the moist primitive equations. To overcome the difficulty, we try to use Aubin-Lions lemma and the continuous property of the strong solutions to the moist primitive equations to prove the the existence of the global attractor which improves the result, the existence of weak attractor, in $\cite{GH2}$.