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.

[2]

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

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[35]

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

[36]

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

[37]

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

[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.

[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.

[40]

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

[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.

[42]

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

[43]

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

[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.

[45]

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

[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.

[47]

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

[48]

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

[49]

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

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[2]

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

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[35]

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

[36]

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

[37]

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

[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.

[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.

[40]

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

[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.

[42]

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

[43]

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

[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.

[45]

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

[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.

[47]

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

[48]

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

[49]

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

[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.

[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.

[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.

[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.

[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.

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