March  2022, 42(3): 1535-1568. doi: 10.3934/dcds.2021163

On long-time asymptotic behavior for solutions to 2D temperature-dependent tropical climate model

1. 

Department of Mathematics, Changzhou University, Changzhou 213164, Jiangsu, China

2. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

3. 

Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, China

* Corresponding author: Zhuan Ye

Received  February 2021 Revised  August 2021 Published  March 2022 Early access  November 2021

Fund Project: Xu was partially supported by the National Natural Science Foundation of China (Grants No. 12171040, No. 11771045 and No.11871087) and the National Key R & D Programe of China(Grant No. 2020YFA0712900). Ye is supported by the National Natural Science Foundation of China (Grant No. 11701232), the Natural Science Foundation of Jiangsu Province (No. BK20170224) and the Qing Lan Project of Jiangsu Province

In this paper, we are concerned with the long-time asymptotic behavior of the two-dimensional temperature-dependent tropical climate model. More precisely, we obtain the sharp time-decay of the solution of the system with the general initial data belonging to an appropriate Sobolev space with negative indices. In addition, when such condition of the initial data is absent, it is shown that any spatial derivative of the positive integer $ k $-order of the solution actually decays at least at the rate of $ (1+t)^{-\frac{k}{2}} $.

Citation: Chaoying Li, Xiaojing Xu, Zhuan Ye. On long-time asymptotic behavior for solutions to 2D temperature-dependent tropical climate model. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1535-1568. doi: 10.3934/dcds.2021163
References:
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R. Agapito and M. Schonbek, Non-uniform decay of MHD equations with and without magnetic diffusion, Comm. Partial Differential Equations, 32 (2007), 1791-1812.  doi: 10.1080/03605300701318658.

[2]

Q. Chen and L. Jiang, Global well-posedness for the 2-D Boussinesq system with temperature-dependent thermal diffusivity, Colloq. Math., 135 (2014), 187-199.  doi: 10.4064/cm135-2-3.

[3]

B.-Q. DongC. LiX. Xu and Z. Ye, Global smooth solution of 2D temperature-dependent tropical climate model, Nonlinearity, 34 (2021), 5662-5686.  doi: 10.1088/1361-6544/ac0d44.

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B.-Q. DongW. WangJ. WuZ. Ye and H. Zhang, Global regularity for a class of 2D generalized tropical climate models, J. Differential Equations, 266 (2019), 6346-6382.  doi: 10.1016/j.jde.2018.11.007.

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B. DongW. WangJ. Wu and H. Zhang, Global regularity results for the climate model with fractional dissipation, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 211-229.  doi: 10.3934/dcdsb.2018102.

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B.-Q. DongJ. Wu and Z. Ye, Global regularity for a 2D tropical climate model with fractional dissipation, J. Nonlinear Sci., 29 (2019), 511-550.  doi: 10.1007/s00332-018-9495-5.

[7]

B.-Q. DongJ. Wu and Z. Ye, 2D tropical climate model with fractional dissipation and without thermal diffusion, Commun. Math. Sci., 18 (2020), 259-292.  doi: 10.4310/CMS.2020.v18.n1.a11.

[8]

D. M. W. FriersonA. J. Majda and O. M. Pauluis, Large scale dynamics of precipitation fronts in the tropical atmosphere: A novel relaxation limit, Commun. Math. Sci., 2 (2004), 591-626.  doi: 10.4310/CMS.2004.v2.n4.a3.

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A. E. Gill, Some simple solutions for heat-induced tropical circulation, Quart. J. Roy. Meteor. Soc., 106 (1980), 447-462.  doi: 10.1002/qj.49710644905.

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T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[11]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[12]

B. Khouider and A. J. Majda, A non–oscillatory well balanced scheme for an idealized tropical climate model: Ⅰ. Algorithm and validation, Theor. Comput. Fluid Dyn., 19 (2005), 331-354.  doi: 10.1007/s00162-005-0170-8.

[13]

D. Li and X. Xu, Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity, Dyn. Partial Differ. Equ., 10 (2013), 255-265.  doi: 10.4310/DPDE.2013.v10.n3.a2.

[14]

H. Li and Y. Xiao, Decay rate of unique global solution for a class of 2D tropical climate model, Math. Methods Appl. Sci., 42 (2019), 2533-2543.  doi: 10.1002/mma.5529.

[15]

J. Li and E. Titi, Global well-posedness of strong solutions to a tropical climate model, Discrete Contin. Dyn. Syst., 36 (2016), 4495-4516.  doi: 10.3934/dcds.2016.36.4495.

[16]

J. Li and E. S. Titi, A tropical atmosphere model with moisture: Global well-posedness and relaxation limit, Nonlinearity, 29 (2016), 2674-2714.  doi: 10.1088/0951-7715/29/9/2674.

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S. A. Lorca and J. L. Boldrini, Stationary solutions for generalized Boussinesq models, J. Differential Equations, 124 (1996), 389-406.  doi: 10.1006/jdeq.1996.0016.

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A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9. American Mathematical Society, Providence 2003. doi: 10.1090/cln/009.

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A. J. Majda and J. A. Biello, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmos. Sci., 60 (2003), 1809-1821.  doi: 10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2.

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K. Masuda, Weak solutions of the Navier–Stokes equations, Tohoku Math. J., 36 (1984), 623-646.  doi: 10.2748/tmj/1178228767.

[21]

T. Matsuno, Quasi-geostrophic motions in the equatorial area, J. Meteor. Soc. Japan, 44 (1966), 25-43.  doi: 10.2151/jmsj1965.44.1_25.

[22]

T. OgawaS. V. Rajopadhye and M. E. Schonbek, Energy decay for a weak solution of the Navier-Stokes equation with slowly varying external forces, J. Funct. Anal., 144 (1997), 325-358.  doi: 10.1006/jfan.1996.3011.

[23]

J. T. RatcliffP. J. TackleyG. Schubert and A. Zebib, Transitions in thermal convection with strongly variable viscosity, Phys. Earth Planet. Inter., 102 (1997), 201-212.  doi: 10.1016/S0031-9201(97)00013-7.

[24]

M. E. Schonbek, $L^2$ decay for weak solutions of the Navier–Stokes equations, Arch. Rational Mech. Anal., 88 (1985), 209-222.  doi: 10.1007/BF00752111.

[25]

M. E. Schonbek, Large time behavior of solutions to Navier–Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443.

[26]

M. E. Schonbek and M. Wiegner, On the decay of higher-order norms of the solutions of Navier-Stokes equations, Proc. R. Soc. Edinb., Sect. A, 126 (1996), 677-685.  doi: 10.1017/S0308210500022976.

[27]

S. N. Stechmann and A. J. Majda, The structure of precipitation fronts for finite relaxation time, Theor. Comput. Fluid Dyn., 20 (2006), 377-404.  doi: 10.1007/s00162-006-0014-1.

[28]

Y. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085.  doi: 10.1016/j.jde.2013.04.032.

[29]

H. Triebel, Theory of Function Spaces, Monogr. Math., Birkh$\rm\ddot{a}$user Verlag, Basel, Boston, 1983. doi: 10.1007/978-3-0346-0416-1.

[30]

D. Turcotte and G. Schubert, Geodynamics Applications of Continuum Physics to Geological Problems, John Wiley and Sons, 1982.

[31]

R. Wan, Global small solutions to a tropical climate model without thermal diffusion, J. Math. Phys., 57 (2016), 021507, 13 pp. doi: 10.1063/1.4941039.

[32]

C. Wang and Z. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math., 228 (2011), 43-62.  doi: 10.1016/j.aim.2011.05.008.

[33]

X. Ye and M. Zhu, Global strong solutions of the 2D tropical climate system with temperature-dependent viscosity, Z. Angew. Math. Phys., 71 (2020), Paper No. 97, 10 pp. doi: 10.1007/s00033-020-01321-9.

[34]

Z. Ye, Global regularity for a class of 2D tropical climate model, J. Math. Anal. Appl., 446 (2017), 307-321.  doi: 10.1016/j.jmaa.2016.08.053.

[35]

Z. Ye, Global regularity of 2D tropical climate model with zero thermal diffusion, ZAMM Z. Angew. Math. Mech., 100 (2020), e201900132, 20 pp. doi: 10.1002/zamm.201900132.

[36]

X. Zhai and Y. Chen, Global strong solutions and time decay of 2D tropical climate model with zero thermal diffusion, Math. Methods Appl. Sci., 43 (2020), 7022-7039.  doi: 10.1002/mma.6452.

show all references

References:
[1]

R. Agapito and M. Schonbek, Non-uniform decay of MHD equations with and without magnetic diffusion, Comm. Partial Differential Equations, 32 (2007), 1791-1812.  doi: 10.1080/03605300701318658.

[2]

Q. Chen and L. Jiang, Global well-posedness for the 2-D Boussinesq system with temperature-dependent thermal diffusivity, Colloq. Math., 135 (2014), 187-199.  doi: 10.4064/cm135-2-3.

[3]

B.-Q. DongC. LiX. Xu and Z. Ye, Global smooth solution of 2D temperature-dependent tropical climate model, Nonlinearity, 34 (2021), 5662-5686.  doi: 10.1088/1361-6544/ac0d44.

[4]

B.-Q. DongW. WangJ. WuZ. Ye and H. Zhang, Global regularity for a class of 2D generalized tropical climate models, J. Differential Equations, 266 (2019), 6346-6382.  doi: 10.1016/j.jde.2018.11.007.

[5]

B. DongW. WangJ. Wu and H. Zhang, Global regularity results for the climate model with fractional dissipation, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 211-229.  doi: 10.3934/dcdsb.2018102.

[6]

B.-Q. DongJ. Wu and Z. Ye, Global regularity for a 2D tropical climate model with fractional dissipation, J. Nonlinear Sci., 29 (2019), 511-550.  doi: 10.1007/s00332-018-9495-5.

[7]

B.-Q. DongJ. Wu and Z. Ye, 2D tropical climate model with fractional dissipation and without thermal diffusion, Commun. Math. Sci., 18 (2020), 259-292.  doi: 10.4310/CMS.2020.v18.n1.a11.

[8]

D. M. W. FriersonA. J. Majda and O. M. Pauluis, Large scale dynamics of precipitation fronts in the tropical atmosphere: A novel relaxation limit, Commun. Math. Sci., 2 (2004), 591-626.  doi: 10.4310/CMS.2004.v2.n4.a3.

[9]

A. E. Gill, Some simple solutions for heat-induced tropical circulation, Quart. J. Roy. Meteor. Soc., 106 (1980), 447-462.  doi: 10.1002/qj.49710644905.

[10]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[11]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[12]

B. Khouider and A. J. Majda, A non–oscillatory well balanced scheme for an idealized tropical climate model: Ⅰ. Algorithm and validation, Theor. Comput. Fluid Dyn., 19 (2005), 331-354.  doi: 10.1007/s00162-005-0170-8.

[13]

D. Li and X. Xu, Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity, Dyn. Partial Differ. Equ., 10 (2013), 255-265.  doi: 10.4310/DPDE.2013.v10.n3.a2.

[14]

H. Li and Y. Xiao, Decay rate of unique global solution for a class of 2D tropical climate model, Math. Methods Appl. Sci., 42 (2019), 2533-2543.  doi: 10.1002/mma.5529.

[15]

J. Li and E. Titi, Global well-posedness of strong solutions to a tropical climate model, Discrete Contin. Dyn. Syst., 36 (2016), 4495-4516.  doi: 10.3934/dcds.2016.36.4495.

[16]

J. Li and E. S. Titi, A tropical atmosphere model with moisture: Global well-posedness and relaxation limit, Nonlinearity, 29 (2016), 2674-2714.  doi: 10.1088/0951-7715/29/9/2674.

[17]

S. A. Lorca and J. L. Boldrini, Stationary solutions for generalized Boussinesq models, J. Differential Equations, 124 (1996), 389-406.  doi: 10.1006/jdeq.1996.0016.

[18]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9. American Mathematical Society, Providence 2003. doi: 10.1090/cln/009.

[19]

A. J. Majda and J. A. Biello, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmos. Sci., 60 (2003), 1809-1821.  doi: 10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2.

[20]

K. Masuda, Weak solutions of the Navier–Stokes equations, Tohoku Math. J., 36 (1984), 623-646.  doi: 10.2748/tmj/1178228767.

[21]

T. Matsuno, Quasi-geostrophic motions in the equatorial area, J. Meteor. Soc. Japan, 44 (1966), 25-43.  doi: 10.2151/jmsj1965.44.1_25.

[22]

T. OgawaS. V. Rajopadhye and M. E. Schonbek, Energy decay for a weak solution of the Navier-Stokes equation with slowly varying external forces, J. Funct. Anal., 144 (1997), 325-358.  doi: 10.1006/jfan.1996.3011.

[23]

J. T. RatcliffP. J. TackleyG. Schubert and A. Zebib, Transitions in thermal convection with strongly variable viscosity, Phys. Earth Planet. Inter., 102 (1997), 201-212.  doi: 10.1016/S0031-9201(97)00013-7.

[24]

M. E. Schonbek, $L^2$ decay for weak solutions of the Navier–Stokes equations, Arch. Rational Mech. Anal., 88 (1985), 209-222.  doi: 10.1007/BF00752111.

[25]

M. E. Schonbek, Large time behavior of solutions to Navier–Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.  doi: 10.1080/03605308608820443.

[26]

M. E. Schonbek and M. Wiegner, On the decay of higher-order norms of the solutions of Navier-Stokes equations, Proc. R. Soc. Edinb., Sect. A, 126 (1996), 677-685.  doi: 10.1017/S0308210500022976.

[27]

S. N. Stechmann and A. J. Majda, The structure of precipitation fronts for finite relaxation time, Theor. Comput. Fluid Dyn., 20 (2006), 377-404.  doi: 10.1007/s00162-006-0014-1.

[28]

Y. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085.  doi: 10.1016/j.jde.2013.04.032.

[29]

H. Triebel, Theory of Function Spaces, Monogr. Math., Birkh$\rm\ddot{a}$user Verlag, Basel, Boston, 1983. doi: 10.1007/978-3-0346-0416-1.

[30]

D. Turcotte and G. Schubert, Geodynamics Applications of Continuum Physics to Geological Problems, John Wiley and Sons, 1982.

[31]

R. Wan, Global small solutions to a tropical climate model without thermal diffusion, J. Math. Phys., 57 (2016), 021507, 13 pp. doi: 10.1063/1.4941039.

[32]

C. Wang and Z. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math., 228 (2011), 43-62.  doi: 10.1016/j.aim.2011.05.008.

[33]

X. Ye and M. Zhu, Global strong solutions of the 2D tropical climate system with temperature-dependent viscosity, Z. Angew. Math. Phys., 71 (2020), Paper No. 97, 10 pp. doi: 10.1007/s00033-020-01321-9.

[34]

Z. Ye, Global regularity for a class of 2D tropical climate model, J. Math. Anal. Appl., 446 (2017), 307-321.  doi: 10.1016/j.jmaa.2016.08.053.

[35]

Z. Ye, Global regularity of 2D tropical climate model with zero thermal diffusion, ZAMM Z. Angew. Math. Mech., 100 (2020), e201900132, 20 pp. doi: 10.1002/zamm.201900132.

[36]

X. Zhai and Y. Chen, Global strong solutions and time decay of 2D tropical climate model with zero thermal diffusion, Math. Methods Appl. Sci., 43 (2020), 7022-7039.  doi: 10.1002/mma.6452.

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