Abstract
For a set of non-periodic boundary conditions, we prove the uniform boundedness of the \(H^2\) norms of the solutions of the 3D Primitive Equations with viscosity. An absorbing set of the solutions in \(H^2\) is also obtained. As an application of this result, we prove also the finiteness of the Hausdorff and fractal dimensions of the global attractor for the strong solutions of the 3D Primitive Equations with viscosity. Our results also improve the existing results for the case with periodic boundary conditions.
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Acknowledgments
This work has benefited from the first author’s research visit in July of 2013 at the Institute of Scientific Computing and Applied Mathematics of Indiana University, Bloomington, Indiana. The first author expresses his gratitude for the financial support by the Department of Mathematics of Indiana University and for the hospitality of the Institute of Scientific Computing and Applied Mathematics. This work was also supported in part by NSF-DMS Grant 1206438 and by the Research Fund of Indiana University.
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Communicated by Edriss S. Titi.
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Ju, N., Temam, R. Finite Dimensions of the Global Attractor for 3D Primitive Equations with Viscosity. J Nonlinear Sci 25, 131–155 (2015). https://doi.org/10.1007/s00332-014-9223-8
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DOI: https://doi.org/10.1007/s00332-014-9223-8