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
References:
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J. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044. Google Scholar

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V. Bjerknes, Das Problem der Wettervorhersage, betrachtet vom Standpunkte der Mechanik und der Physik, Meteorol. Z., 21 (1904), 1-7. Google Scholar

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A. J. Bourgeois and J. T. Beale, Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean, SIAM J. Math. Anal., 25 (1994), 1023-1068. doi: 10.1137/S0036141092234980. Google Scholar

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D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. of Math., 148 (1998), 1135-1152. doi: 10.2307/121037. Google Scholar

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C. CaoS. IbrahimK. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482. doi: 10.1007/s00220-015-2365-1. Google Scholar

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C. CaoJ. Li and E. S. Titi, Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Anal. Ration. Mech., 214 (2014), 35-76. doi: 10.1007/s00205-014-0752-y. Google Scholar

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C. CaoJ. Li and E. S. Titi, Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity, J. Differ. Equ., 257 (2014), 4108-4132. doi: 10.1016/j.jde.2014.08.003. Google Scholar

[9]

C. CaoJ. Li and E. S. Titi, Global well-posedness for the 3D primitive equations with only horizontal viscosity and diffusion, Commun. Pure Appl. Math., 69 (2016), 1492-1531. doi: 10.1002/cpa.21576. Google Scholar

[10]

P. ConstantinA. Majda and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533. doi: 10.1088/0951-7715/7/6/001. Google Scholar

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P. ConstantinA. Majda and E. Tabak, Singular front formation in a model for quasigeostrophic flow, Phys. Fluids, 6 (1994), 9-11. doi: 10.1063/1.868050. Google Scholar

[12]

C. Cao and E. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. Math., 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245. Google Scholar

[13]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional primitive equations with partial vertical turbulence mixing heat diffusion, Commun. Math. Phys., 310 (2012), 537-568. doi: 10.1007/s00220-011-1409-4. Google Scholar

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V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. Google Scholar

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P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948. doi: 10.1137/S0036141098337333. Google Scholar

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P. F. Embid and A. J. Majda, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm. Partial Differential Equations, 21 (1996), 619-658. doi: 10.1080/03605309608821200. Google Scholar

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B. Guo and D. Huang, Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics, J. Math. Phys., 47 (2006), 083508, 23pp. doi: 10.1063/1.2245207. Google Scholar

[20]

B. Guo and D. Huang, Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere, J.Differential Equations, 251 (2011), 457-491. doi: 10.1016/j.jde.2011.05.010. Google Scholar

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F. Guillén-GonzáezN. Masmoudi and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the Primitive Equations, Diff. Integral Eq., 14 (2001), 1381-1408. Google Scholar

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J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. Google Scholar

[23]

G. J. Haltiner, Numerical Weather Prediction, New York: Wiley, 1971.Google Scholar

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G. J. Haltiner and R. T. Williams, Numerical Prediction and Dynamic Meteorology, New York: Wiley, 1980.Google Scholar

[25]

J. R. Holton, An Introduction to Dynamic Meteorology, 3rd edition, Academic Press, 1992.Google Scholar

[26]

C. HuR. Temam and M. Ziane, The primitive equations on the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst., 9 (2003), 97-131. Google Scholar

[27]

N. Ju, The global attractor for the solutions to the 3d viscous primitive equations, Discrete and Continuous Dynamical Systems, 17 (2007), 159-179. doi: 10.3934/dcds.2007.17.159. Google Scholar

[28]

N. Ju, Finite dimensionality of the global attractor for 3d primitive equations with viscosity, Discrete Contin. Dyn. Syst., 17 (2007), 159–179, arXiv: 1507.05992. doi: 10.3934/dcds.2007.17.159. Google Scholar

[29]

N. Ju and R. Teman, Finite dimensions of the global attractor for 3d viscous primitive equations with viscosity, J Nonlinear Sci, 25 (2015), 131-155. doi: 10.1007/s00332-014-9223-8. Google Scholar

[30]

G. M. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. doi: 10.1016/j.crma.2006.04.020. Google Scholar

[31]

G. M. Kobelkov, Existence of a solution 'in the large' for ocean dynamics equations, J. Math. Fluid Mech., 9 (2007), 588-610. doi: 10.1007/s00021-006-0228-4. Google Scholar

[32]

I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753. doi: 10.1088/0951-7715/20/12/001. Google Scholar

[33]

O. Ladyzhenskaya, Some comments to my papers on the theory of attractors for abstract semigroups, Zap. Nauchn. Sem. LOMI, 182 (1990), 102-112. doi: 10.1007/BF01671002. Google Scholar

[34]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418. Google Scholar

[35]

J. Lions, Quelques Méthode de résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. Google Scholar

[36]

J. Li and J. Chou, Asymptotic behavior of solutions of the moist atmospheric equations, Acta Meteor. Sinica, 56 (1998), 61–72 (in Chinese).Google Scholar

[37]

J. Lions and B. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer–Verlag, New York, 1972. Google Scholar

[38]

J. L. LionsR. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288. doi: 10.1088/0951-7715/5/2/001. Google Scholar

[39]

J.L. LionsR. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002. Google Scholar

[40]

J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I), Comput. Mech. Adv., 1 (1993), 120pp. Google Scholar

[41]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lect. Notes Math., vol. 9, 2003. doi: 10.1090/cln/009. Google Scholar

[42]

A. Miranville and R. Temam, Mathematical Modeling in Continuum Mechanics, Cambridge: Cambridge University Press, 2005. doi: 10.1017/CBO9780511755422. Google Scholar

[43]

J. Pedlosky, Geophysical Fluid Dynamics, 2nd edition, Springer-Verlag, Berlin/New York, 1987.Google Scholar

[44]

M. Petcu, R. M. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of Numerical Analysis, 14 Special vol Computational Methods for the Atmosphere and the Oceans, (Amsterdam: Elsevier/North-Holland), (2009), 577–750. doi: 10.1016/S1570-8659(08)00212-3. Google Scholar

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L. F. Richardson, Weather Prediction by Numerical Process, Cambridge Mathematical Library, Cambridge: Cambridge University, 2007. doi: 10.1017/CBO9780511618291. Google Scholar

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J. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar

[47]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. Google Scholar

[48]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringVerlag, 1988, 2nd Edition, 1997. doi: 10.1007/978-1-4684-0313-8. Google Scholar

[49]

R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of Mathematical Fluid Dynamics, 3 (2004), 535-657. Google Scholar

[50]

S. Wang, Attractors for the 3-D baroclinic quasi-geostrophic equations of large-scale atmosphere, J. Math. Anal. Appl., 165 (1992), 266-283. doi: 10.1016/0022-247X(92)90078-R. Google Scholar

[51]

J. Wang, Global solutions of the 2D dissipative quasi-geostrophic equations in Besov spaces, SIAM J. Math. Anal., 36 (2004), 1014-1030. doi: 10.1137/S0036141003435576. Google Scholar

[52]

J. Wang, The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation, Nonlinearity, 18 (2005), 139-154. doi: 10.1088/0951-7715/18/1/008. Google Scholar

[53]

M. C. ZelatiA. HuangL. KukavicaR. Teman and M. Ziane, The primitive equations of the atmosphere in presence of vapour saturation, Nonlinearlity, 28 (2015), 625-668. doi: 10.1088/0951-7715/28/3/625. Google Scholar

[54]

G. L. Zhou and B. L. Guo, The global attractor for the 3-D viscous primitive equations of large-scale moist atmosphere, preprint, arXiv: submit/2262284.Google Scholar

show all references

References:
[1]

J. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044. Google Scholar

[2]

V. Bjerknes, Das Problem der Wettervorhersage, betrachtet vom Standpunkte der Mechanik und der Physik, Meteorol. Z., 21 (1904), 1-7. Google Scholar

[3]

A. J. Bourgeois and J. T. Beale, Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean, SIAM J. Math. Anal., 25 (1994), 1023-1068. doi: 10.1137/S0036141092234980. Google Scholar

[4]

A. Babin and M. Vishik, Attractor of Evolution Equations, North-Holland, Amsterdam, 1992. Google Scholar

[5]

D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. of Math., 148 (1998), 1135-1152. doi: 10.2307/121037. Google Scholar

[6]

C. CaoS. IbrahimK. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482. doi: 10.1007/s00220-015-2365-1. Google Scholar

[7]

C. CaoJ. Li and E. S. Titi, Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Anal. Ration. Mech., 214 (2014), 35-76. doi: 10.1007/s00205-014-0752-y. Google Scholar

[8]

C. CaoJ. Li and E. S. Titi, Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity, J. Differ. Equ., 257 (2014), 4108-4132. doi: 10.1016/j.jde.2014.08.003. Google Scholar

[9]

C. CaoJ. Li and E. S. Titi, Global well-posedness for the 3D primitive equations with only horizontal viscosity and diffusion, Commun. Pure Appl. Math., 69 (2016), 1492-1531. doi: 10.1002/cpa.21576. Google Scholar

[10]

P. ConstantinA. Majda and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533. doi: 10.1088/0951-7715/7/6/001. Google Scholar

[11]

P. ConstantinA. Majda and E. Tabak, Singular front formation in a model for quasigeostrophic flow, Phys. Fluids, 6 (1994), 9-11. doi: 10.1063/1.868050. Google Scholar

[12]

C. Cao and E. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. Math., 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245. Google Scholar

[13]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional primitive equations with partial vertical turbulence mixing heat diffusion, Commun. Math. Phys., 310 (2012), 537-568. doi: 10.1007/s00220-011-1409-4. Google Scholar

[14]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. Google Scholar

[15]

P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948. doi: 10.1137/S0036141098337333. Google Scholar

[16]

P. F. Embid and A. J. Majda, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm. Partial Differential Equations, 21 (1996), 619-658. doi: 10.1080/03605309608821200. Google Scholar

[17]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34. Google Scholar

[18]

A. E. Gill, Atmosphere–Ocean Dynamics, (International Geophysics Series vol 30) (San Diego, CA: Academic), 1982.Google Scholar

[19]

B. Guo and D. Huang, Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics, J. Math. Phys., 47 (2006), 083508, 23pp. doi: 10.1063/1.2245207. Google Scholar

[20]

B. Guo and D. Huang, Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere, J.Differential Equations, 251 (2011), 457-491. doi: 10.1016/j.jde.2011.05.010. Google Scholar

[21]

F. Guillén-GonzáezN. Masmoudi and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the Primitive Equations, Diff. Integral Eq., 14 (2001), 1381-1408. Google Scholar

[22]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. Google Scholar

[23]

G. J. Haltiner, Numerical Weather Prediction, New York: Wiley, 1971.Google Scholar

[24]

G. J. Haltiner and R. T. Williams, Numerical Prediction and Dynamic Meteorology, New York: Wiley, 1980.Google Scholar

[25]

J. R. Holton, An Introduction to Dynamic Meteorology, 3rd edition, Academic Press, 1992.Google Scholar

[26]

C. HuR. Temam and M. Ziane, The primitive equations on the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst., 9 (2003), 97-131. Google Scholar

[27]

N. Ju, The global attractor for the solutions to the 3d viscous primitive equations, Discrete and Continuous Dynamical Systems, 17 (2007), 159-179. doi: 10.3934/dcds.2007.17.159. Google Scholar

[28]

N. Ju, Finite dimensionality of the global attractor for 3d primitive equations with viscosity, Discrete Contin. Dyn. Syst., 17 (2007), 159–179, arXiv: 1507.05992. doi: 10.3934/dcds.2007.17.159. Google Scholar

[29]

N. Ju and R. Teman, Finite dimensions of the global attractor for 3d viscous primitive equations with viscosity, J Nonlinear Sci, 25 (2015), 131-155. doi: 10.1007/s00332-014-9223-8. Google Scholar

[30]

G. M. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. doi: 10.1016/j.crma.2006.04.020. Google Scholar

[31]

G. M. Kobelkov, Existence of a solution 'in the large' for ocean dynamics equations, J. Math. Fluid Mech., 9 (2007), 588-610. doi: 10.1007/s00021-006-0228-4. Google Scholar

[32]

I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753. doi: 10.1088/0951-7715/20/12/001. Google Scholar

[33]

O. Ladyzhenskaya, Some comments to my papers on the theory of attractors for abstract semigroups, Zap. Nauchn. Sem. LOMI, 182 (1990), 102-112. doi: 10.1007/BF01671002. Google Scholar

[34]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418. Google Scholar

[35]

J. Lions, Quelques Méthode de résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. Google Scholar

[36]

J. Li and J. Chou, Asymptotic behavior of solutions of the moist atmospheric equations, Acta Meteor. Sinica, 56 (1998), 61–72 (in Chinese).Google Scholar

[37]

J. Lions and B. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer–Verlag, New York, 1972. Google Scholar

[38]

J. L. LionsR. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288. doi: 10.1088/0951-7715/5/2/001. Google Scholar

[39]

J.L. LionsR. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002. Google Scholar

[40]

J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I), Comput. Mech. Adv., 1 (1993), 120pp. Google Scholar

[41]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lect. Notes Math., vol. 9, 2003. doi: 10.1090/cln/009. Google Scholar

[42]

A. Miranville and R. Temam, Mathematical Modeling in Continuum Mechanics, Cambridge: Cambridge University Press, 2005. doi: 10.1017/CBO9780511755422. Google Scholar

[43]

J. Pedlosky, Geophysical Fluid Dynamics, 2nd edition, Springer-Verlag, Berlin/New York, 1987.Google Scholar

[44]

M. Petcu, R. M. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of Numerical Analysis, 14 Special vol Computational Methods for the Atmosphere and the Oceans, (Amsterdam: Elsevier/North-Holland), (2009), 577–750. doi: 10.1016/S1570-8659(08)00212-3. Google Scholar

[45]

L. F. Richardson, Weather Prediction by Numerical Process, Cambridge Mathematical Library, Cambridge: Cambridge University, 2007. doi: 10.1017/CBO9780511618291. Google Scholar

[46]

J. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar

[47]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. Google Scholar

[48]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringVerlag, 1988, 2nd Edition, 1997. doi: 10.1007/978-1-4684-0313-8. Google Scholar

[49]

R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of Mathematical Fluid Dynamics, 3 (2004), 535-657. Google Scholar

[50]

S. Wang, Attractors for the 3-D baroclinic quasi-geostrophic equations of large-scale atmosphere, J. Math. Anal. Appl., 165 (1992), 266-283. doi: 10.1016/0022-247X(92)90078-R. Google Scholar

[51]

J. Wang, Global solutions of the 2D dissipative quasi-geostrophic equations in Besov spaces, SIAM J. Math. Anal., 36 (2004), 1014-1030. doi: 10.1137/S0036141003435576. Google Scholar

[52]

J. Wang, The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation, Nonlinearity, 18 (2005), 139-154. doi: 10.1088/0951-7715/18/1/008. Google Scholar

[53]

M. C. ZelatiA. HuangL. KukavicaR. Teman and M. Ziane, The primitive equations of the atmosphere in presence of vapour saturation, Nonlinearlity, 28 (2015), 625-668. doi: 10.1088/0951-7715/28/3/625. Google Scholar

[54]

G. L. Zhou and B. L. Guo, The global attractor for the 3-D viscous primitive equations of large-scale moist atmosphere, preprint, arXiv: submit/2262284.Google Scholar

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