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August  2016, 36(8): 4495-4516. doi: 10.3934/dcds.2016.36.4495

Global well-posedness of strong solutions to a tropical climate model

Jinkai Li 1, and  Edriss Titi 2,

1. 

Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
2. 

Department of Mathematics, Texas A&M University, 3368-TAMU, College Station, TX 77843-3368, United States

Received  April 2015 Revised  October 2015 Published  March 2016

In this paper, we consider the Cauchy problem to the TROPICAL CLIMATE MODEL derived by Frierson--Majda--Pauluis in [15], which is a coupled system of the barotropic and baroclinic modes of the velocity and the typical midtropospheric temperature. The system considered in this paper has viscosities in the momentum equations, but no diffusivity in the temperature equation. We establish here the global well-posedness of strong solutions to this model. In proving the global existence of strong solutions, to overcome the difficulty caused by the absence of the diffusivity in the temperature equation, we introduce a new velocity $w$ (called the pseudo baroclinic velocity), which has more regularities than the original baroclinic mode of the velocity. An auxiliary function $\phi$, which looks like the effective viscous flux for the compressible Navier-Stokes equations, is also introduced to obtain the $L^\infty$ bound of the temperature. Regarding the uniqueness, we use the idea of performing suitable energy estimates at level one order lower than the natural basic energy estimates for the system.
Keywords: strong solutions, primitive equations, global well-posedness, Tropical atmospheric dynamics, logarithmic Gronwall inequality..
Mathematics Subject Classification: Primary: 35D35, 76D03; Secondary: 86A1.
Citation: Jinkai Li, Edriss Titi. Global well-posedness of strong solutions to a tropical climate model. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4495-4516. doi: 10.3934/dcds.2016.36.4495
References:
[1]

H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.

[2]

H. Brézis and S. Wainger, A Note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.

[3]

C. Cao, S. Ibrahim, K. 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.

[4]

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

[5]

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

[6]

C. Cao, J. Li and E. S. Titi, Global well-posedness of the 3D primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl. Math., arXiv:1406.1995v1. doi: 10.1002/cpa.21576.

[7]

C. Cao, J. Li and E. S. Titi, Strong solutions to the 3D primitive equations with horizontal dissipation: near $H^1$ initial data, preprint.

[8]

C. Cao, J. Li and E. S. Titi, Global well-posedness of the 3D primitive equations with horizontal viscosities and vertical diffusion, preprint.

[9]

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

[10]

C. Cao and E. S. Titi, Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion, Comm. Math. Phys., 310 (2012), 537-568. doi: 10.1007/s00220-011-1409-4.

[11]

R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635. doi: 10.2307/1970954.

[12]

R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331. doi: 10.1090/S0002-9947-1975-0380244-8.

[13]

L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[14]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.

[15]

D. M. W. Frierson, A. 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.

[16]

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.

[17]

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.

[18]

I. Kukavica and M. Ziane, The regularity of solutions of the primitive equations of the ocean in space dimension three, C. R. Math. Acad. Sci. Paris, 345 (2007), 257-260. doi: 10.1016/j.crma.2007.07.025.

[19]

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.

[20]

A. Larios, E. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations, 255 (2013), 2636-2654. doi: 10.1016/j.jde.2013.07.011.

[21]

J. Li and E. S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 220 (2016), 983-1001, arXiv:1502.06180. doi: 10.1007/s00205-015-0946-y.

[22]

J. Li, E. S. Titi and Z. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb R^2$, Math. Models Methods Appl. Sci., 26 (2016), 803-822, arXiv:1410.1119. doi: 10.1142/S0218202516500184.

[23]

J. L. Lions, R. 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.

[24]

J. L. Lions, R. 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.

[25]

J. L. Lions, R. Temam and S. Wang, Mathematical theory for the coupled atmosphere-ocean models (CAO III), J. Math. Pures Appl., 74 (1995), 105-163.

[26]

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.

[27]

T. Matsuno, Quasi-geostrophic motions in the equatorial area, J. Meteor. Soc. Japan, 44 (1966), 25-42.

[28]

T. K. Wong, Blowup of solutions of the hydrostatic Euler equations, Proc. Amer. Math. Soc., 143 (2015), 1119-1125. doi: 10.1090/S0002-9939-2014-12243-X.

show all references

References:
[1]

H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.

[2]

H. Brézis and S. Wainger, A Note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.

[3]

C. Cao, S. Ibrahim, K. 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.

[4]

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

[5]

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

[6]

C. Cao, J. Li and E. S. Titi, Global well-posedness of the 3D primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl. Math., arXiv:1406.1995v1. doi: 10.1002/cpa.21576.

[7]

C. Cao, J. Li and E. S. Titi, Strong solutions to the 3D primitive equations with horizontal dissipation: near $H^1$ initial data, preprint.

[8]

C. Cao, J. Li and E. S. Titi, Global well-posedness of the 3D primitive equations with horizontal viscosities and vertical diffusion, preprint.

[9]

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

[10]

C. Cao and E. S. Titi, Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion, Comm. Math. Phys., 310 (2012), 537-568. doi: 10.1007/s00220-011-1409-4.

[11]

R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635. doi: 10.2307/1970954.

[12]

R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331. doi: 10.1090/S0002-9947-1975-0380244-8.

[13]

L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[14]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.

[15]

D. M. W. Frierson, A. 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.

[16]

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.

[17]

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.

[18]

I. Kukavica and M. Ziane, The regularity of solutions of the primitive equations of the ocean in space dimension three, C. R. Math. Acad. Sci. Paris, 345 (2007), 257-260. doi: 10.1016/j.crma.2007.07.025.

[19]

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.

[20]

A. Larios, E. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations, 255 (2013), 2636-2654. doi: 10.1016/j.jde.2013.07.011.

[21]

J. Li and E. S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 220 (2016), 983-1001, arXiv:1502.06180. doi: 10.1007/s00205-015-0946-y.

[22]

J. Li, E. S. Titi and Z. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb R^2$, Math. Models Methods Appl. Sci., 26 (2016), 803-822, arXiv:1410.1119. doi: 10.1142/S0218202516500184.

[23]

J. L. Lions, R. 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.

[24]

J. L. Lions, R. 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.

[25]

J. L. Lions, R. Temam and S. Wang, Mathematical theory for the coupled atmosphere-ocean models (CAO III), J. Math. Pures Appl., 74 (1995), 105-163.

[26]

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.

[27]

T. Matsuno, Quasi-geostrophic motions in the equatorial area, J. Meteor. Soc. Japan, 44 (1966), 25-42.

[28]

T. K. Wong, Blowup of solutions of the hydrostatic Euler equations, Proc. Amer. Math. Soc., 143 (2015), 1119-1125. doi: 10.1090/S0002-9939-2014-12243-X.

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