American Institute of Mathematical Sciences

December  2018, 23(10): 4305-4327. doi: 10.3934/dcdsb.2018160

Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere

 1 No 6, Huayuan Road, Haidian District, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, MO 100088, China 2 No 174, Shazhengjie Street, Shapingba District, Chongqing University, Chongqing, MO 401331, China

* Corresponding author: Guoli Zhou*

Received  January 2017 Revised  March 2018 Published  June 2018

Fund Project: The corresponding author was partially supported by NNSF of China(Grant No. 11401057), Natural Science Foundation Project of CQ (Grant No. cstc2016jcyjA0326), Fundamental Research Funds for the Central Universities(Grant No. 106112015CDJXY100005) and China Scholarship Council (Grant No.201506055003)

Under general boundary conditions we consider the finiteness of the Hausdorff and fractal dimensions of the global attractor for the strong solution of the 3D moist primitive equations with viscosity. Firstly, we obtain time-uniform estimates of the first-order time derivative of the strong solutions in $L^2(\mho)$. Then, to prove the finiteness of the Hausdorff and fractal dimensions of the global attractor, the common method is to obtain the uniform boundedness of the strong solution in $H^2(\mho)$ to establish the squeezing property of the solution operator. But it is difficult to achieve due to the boundary conditions and complicated structure of the 3D moist primitive equations. To overcome the difficulties, we try to use the uniform boundedness of the derivative of the strong solutions with respect to time $t$ in $L^2(\mho)$ to prove the uniform continuity of the global attractor. Finally, using the uniform continuity of the global attractor we establish the squeezing property of the solution operator which implies the finiteness of the Hausdorff and fractal dimensions of the global attractor.

Citation: Boling Guo, Guoli Zhou. Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4305-4327. doi: 10.3934/dcdsb.2018160
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