doi: 10.3934/dcds.2021163
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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 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 & Continuous Dynamical Systems, doi: 10.3934/dcds.2021163
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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