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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().  doi: 10.1002/cpa.21576.  Google Scholar

[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., ().   Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().  doi: 10.1002/cpa.21576.  Google Scholar

[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., ().   Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

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