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December  2016, 36(12): 7001-7020. doi: 10.3934/dcds.2016104

## The finite dimensional global attractor for the 3D viscous Primitive Equations

 1 Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078

Received  January 2016 Revised  April 2016 Published  October 2016

A new method is presented to prove finiteness of the fractal and Hausdorff dimensions of the global attractor for the strong solutions to the 3D Primitive Equations (PEs) with viscosity, which is applicable to more general situations than the recent result of [8] in the sense that it removes all extra technical conditions imposed by previous analyses. More specifically, the dimensions of the global attractor are proved finite for heat source $Q\in L^2$, exactly the same condition for well-posedness of global strong solutions and existence of the global attractor of these solutions; while the best previous result obtained recently in [8] still requires the extra condition that $∂_zQ\in L^2$ for finiteness of the dimensions of the global attractor. The key new idea is that Ladyzhenskaya's squeezing property of the semigroup for the strong solutions can be established without higher solution regularity of Primitive Equations. This has the general interest for dissipative evolution equations. For this reason, the new method especially has the advantange of dealing with more complicated boundary conditions which present essential difficulties for previous methods. In particular, the case of 3D viscous PEs with  physical boundary conditions'' can be treated by the new method in the same way as presented in this article, however, it seems rather difficult for previous methods.
Citation: Ning Ju. The finite dimensional global attractor for the 3D viscous Primitive Equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7001-7020. doi: 10.3934/dcds.2016104
##### References:
 [1] C. Cao and E. S. Titi, Global well-posedness and finite dimensional global attractor for a 3-D planetary geostrophic viscous model,, Comm. Pure Appl. Math., 56 (2003), 198. doi: 10.1002/cpa.10056. Google Scholar [2] C. Cao and E. S. Titi, Global well-posedness of the three-dimensional primitive equations of large scale ocean and atmosphere dynamics,, Ann. of Math.(2), 166 (2007), 245. doi: 10.4007/annals.2007.166.245. Google Scholar [3] I. Chueshov, A squeezing property and its applications to a description of long time behaviour in the 3D viscous primitive equations,, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 711. doi: 10.1017/S0308210512001953. Google Scholar [4] P. Constantin, C. Foias and R. Temam, Attractors representing turbulent flows,, Memoirs of A.M.S., 53 (1985). doi: 10.1090/memo/0314. Google Scholar [5] L. Evans and R. Gastler, Some results for the primitive equations with physical boundary conditions,, Z. Angew. Math. Phys., 64 (2013), 1729. doi: 10.1007/s00033-013-0320-6. Google Scholar [6] F. Guillén-Gonzáez, N. Masmoudi and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations,, Diff. Integral Eq., 14 (2001), 1381. Google Scholar [7] N. Ju, The global attractor for the solutions to the 3D viscous primitive equations,, Discrete and Continuous Dynamical Systems, 17 (2007), 159. doi: 10.3934/dcds.2007.17.159. Google Scholar [8] N. Ju and R. Temam, Finite dimensions of the global attractor for 3D primitive equations with viscosity,, J. Nonlinear Sci., 25 (2015), 131. doi: 10.1007/s00332-014-9223-8. Google Scholar [9] G. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations,, C. R. Math. Acad. Sc. Paris, 343 (2006), 283. doi: 10.1016/j.crma.2006.04.020. Google Scholar [10] G. Kobelkov, Existence of a solution 'in the large' for ocean dynamics equations,, J. Math. Fluid Mech., 9 (2007), 588. doi: 10.1007/s00021-006-0228-4. Google Scholar [11] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean,, Nonlinearity, 20 (2007), 2739. doi: 10.1088/0951-7715/20/12/001. Google Scholar [12] I. Kukavica and M. Ziane, Uniform gradient bounds for the primitive equations of the ocean,, Differential Integral Equations, 21 (2008), 837. Google Scholar [13] O. Ladyzhenskaya, Some comments to my papers on the theory of attractors for abstract semigroups,, Zap. Nauchn. Sem. LOMI, 182 (1990), 102. doi: 10.1007/BF01671002. Google Scholar [14] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications,, Nonlinearity, 5 (1992), 237. doi: 10.1088/0951-7715/5/2/001. Google Scholar [15] J. Lions, R. Temam and S. Wang, On the equations of the large scale Ocean,, Nonlinearity, 5 (1992), 1007. doi: 10.1088/0951-7715/5/5/002. Google Scholar [16] M. Petcu, On the three dimensional primitive equations,, Adv. Dif. Eq., 11 (2006), 1201. Google Scholar [17] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [18] R. Temam, Navier-Stokes equations. Theory and numerical analysis,, reprint of 3rd edition, (2001). doi: 10.1090/chel/343. Google Scholar [19] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics,, Handbook of Mathematical Fluid Dynamics, 3 (2004), 535. Google Scholar

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##### References:
 [1] C. Cao and E. S. Titi, Global well-posedness and finite dimensional global attractor for a 3-D planetary geostrophic viscous model,, Comm. Pure Appl. Math., 56 (2003), 198. doi: 10.1002/cpa.10056. Google Scholar [2] C. Cao and E. S. Titi, Global well-posedness of the three-dimensional primitive equations of large scale ocean and atmosphere dynamics,, Ann. of Math.(2), 166 (2007), 245. doi: 10.4007/annals.2007.166.245. Google Scholar [3] I. Chueshov, A squeezing property and its applications to a description of long time behaviour in the 3D viscous primitive equations,, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 711. doi: 10.1017/S0308210512001953. Google Scholar [4] P. Constantin, C. Foias and R. Temam, Attractors representing turbulent flows,, Memoirs of A.M.S., 53 (1985). doi: 10.1090/memo/0314. Google Scholar [5] L. Evans and R. Gastler, Some results for the primitive equations with physical boundary conditions,, Z. Angew. Math. Phys., 64 (2013), 1729. doi: 10.1007/s00033-013-0320-6. Google Scholar [6] F. Guillén-Gonzáez, N. Masmoudi and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations,, Diff. Integral Eq., 14 (2001), 1381. Google Scholar [7] N. Ju, The global attractor for the solutions to the 3D viscous primitive equations,, Discrete and Continuous Dynamical Systems, 17 (2007), 159. doi: 10.3934/dcds.2007.17.159. Google Scholar [8] N. Ju and R. Temam, Finite dimensions of the global attractor for 3D primitive equations with viscosity,, J. Nonlinear Sci., 25 (2015), 131. doi: 10.1007/s00332-014-9223-8. Google Scholar [9] G. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations,, C. R. Math. Acad. Sc. Paris, 343 (2006), 283. doi: 10.1016/j.crma.2006.04.020. Google Scholar [10] G. Kobelkov, Existence of a solution 'in the large' for ocean dynamics equations,, J. Math. Fluid Mech., 9 (2007), 588. doi: 10.1007/s00021-006-0228-4. Google Scholar [11] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean,, Nonlinearity, 20 (2007), 2739. doi: 10.1088/0951-7715/20/12/001. Google Scholar [12] I. Kukavica and M. Ziane, Uniform gradient bounds for the primitive equations of the ocean,, Differential Integral Equations, 21 (2008), 837. Google Scholar [13] O. Ladyzhenskaya, Some comments to my papers on the theory of attractors for abstract semigroups,, Zap. Nauchn. Sem. LOMI, 182 (1990), 102. doi: 10.1007/BF01671002. Google Scholar [14] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications,, Nonlinearity, 5 (1992), 237. doi: 10.1088/0951-7715/5/2/001. Google Scholar [15] J. Lions, R. Temam and S. Wang, On the equations of the large scale Ocean,, Nonlinearity, 5 (1992), 1007. doi: 10.1088/0951-7715/5/5/002. Google Scholar [16] M. Petcu, On the three dimensional primitive equations,, Adv. Dif. Eq., 11 (2006), 1201. Google Scholar [17] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [18] R. Temam, Navier-Stokes equations. Theory and numerical analysis,, reprint of 3rd edition, (2001). doi: 10.1090/chel/343. Google Scholar [19] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics,, Handbook of Mathematical Fluid Dynamics, 3 (2004), 535. Google Scholar
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