July  2020, 40(7): 4287-4305. doi: 10.3934/dcds.2020181

On weak-strong uniqueness and singular limit for the compressible Primitive Equations

1. 

Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

2. 

Key Laboratory of Ministry of Education for Virtual Geographic Environment, Jiangsu Center for Collaborative Innovation in Geographical, Information Resource Development and Application, Nanjing Normal University, Nanjing 210023, China

3. 

Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 11567, Praha 1, Czech Republic

4. 

Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, China

* Corresponding author

Received  June 2019 Revised  January 2020 Published  April 2020

Fund Project: The research of H. G is partially supported by the NSFC Grant No. 11531006. The research of Š.N. leading to these results has received funding from the Czech Sciences Foundation (GAČR), GA19-04243S and RVO 67985840. The research of T.T. is supported by the NSFC Grant No. 11801138

The paper addresses the weak-strong uniqueness property and singular limit for the compressible Primitive Equations (PE). We show that a weak solution coincides with the strong solution emanating from the same initial data. On the other hand, we prove compressible PE will approach to the incompressible inviscid PE equations in the regime of low Mach number and large Reynolds number in the case of well-prepared initial data. To the best of the authors' knowledge, this is the first work to bridge the link between the compressible PE with incompressible inviscid PE.

Citation: Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181
References:
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Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity, 12 (1999), 495-512.  doi: 10.1088/0951-7715/12/3/004.

[2]

Y. Brenier, Remarks on the derivation of the hydrostatic Euler equations, Bull. Sci. Math., 127 (2003), 585-595.  doi: 10.1016/S0007-4497(03)00024-1.

[3]

D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl., 86 (2006), 362-368.  doi: 10.1016/j.matpur.2006.06.005.

[4]

D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., 87 (2007), 57-90.  doi: 10.1016/j.matpur.2006.11.001.

[5]

D. BreschF. Guillén-GonzálezN. Masmoudi and M. A. Rodríguez-Bellido, On the uniqueness of weak solutions of the two-dimensional primitive equations, Differential Integral Equations, 16 (2003), 77-94. 

[6]

D. Bresch and P.-E. Jabin, Global existence of weak solutions for compressible Navier-Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. of Math., 188 (2018), 577-684.  doi: 10.4007/annals.2018.188.2.4.

[7]

D. BreschA. Kazhikhov and J. Lemoine, On the two-dimensional hydrostatic Navier-Stokes equations, SIAM J. Math. Anal., 36 (2004/05), 796-814.  doi: 10.1137/S0036141003422242.

[8]

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.

[9]

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. Ration. Mech. Anal., 214 (2014), 35-76.  doi: 10.1007/s00205-014-0752-y.

[10]

C. CaoJ. Li and E. S. Titi, Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion, Comm. Pure Appl. Math., 69 (2016), 1492-1531.  doi: 10.1002/cpa.21576.

[11]

C. CaoJ. Li and E. S. Titi, Strong solutions to the 3D primitive equations with only horizontal dissipation: Near $H^1$ initial data, J. Funct. Anal., 272 (2017), 4606-4641.  doi: 10.1016/j.jfa.2017.01.018.

[12]

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.

[13] J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Mathematical Geophysics: An Introduction to Rotating Fluids and the Navier-Stokes Equations, The Clarendon Press, Oxford University Press, Oxford, 2006. 
[14]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.

[15]

A. DebusscheN. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.

[16]

A. DebusscheN. Glatt-HoltzR. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118.  doi: 10.1088/0951-7715/25/7/2093.

[17]

Z. DongJ. Zhai and R. Zhang, Large deviation principles for 3D stochastic primitive equations, J. Differential Equations, 263 (2017), 3110-3146.  doi: 10.1016/j.jde.2017.04.025.

[18]

M. Ersoy and T. Ngom, Existence of a global weak solution to compressible primitive equations, C. R. Math. Acad. Sci. Paris, 350 (2012), 379-382.  doi: 10.1016/j.crma.2012.04.013.

[19]

M. ErsoyT. Ngom and M. Sy, Compressible primitive equations: Formal derivation and stability of weak solutions, Nonlinearity, 24 (2011), 79-96.  doi: 10.1088/0951-7715/24/1/004.

[20]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Vol. 26, Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2004.

[21]

E. FeireislI. Gallagher and A. Novotný, A singular limit for compressible rotating fluids, SIAM J. Math. Anal., 44 (2012), 192-205.  doi: 10.1137/100808010.

[22]

E. FeireislB. J. Jin and A. Novotný, Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech., 14 (2012), 717-730.  doi: 10.1007/s00021-011-0091-9.

[23]

E. FeireislB. J. Jin and A. Novotný, Inviscid incompressible limits of strongly stratified fluids, Asymptot. Anal., 89 (2014), 307-329.  doi: 10.3233/ASY-141231.

[24]

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.

[25]

H. Gao and C. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3053-3073.  doi: 10.3934/dcdsb.2016087.

[26]

B. V. Gatapov and A. V. Kazhikhov, Existence of a global solution of a model problem of atmospheric dynamics, Siberian Math. J., 46 (2005), 805-812.  doi: 10.1007/s11202-005-0079-x.

[27]

P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 13 (2011), 137-146.  doi: 10.1007/s00021-009-0006-1.

[28]

F. Guillén-GonzálezN. Masmoudi and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations, Differential Integral Equations, 14 (2001), 1381-1408. 

[29]

B. Guo and D. Huang, Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics, J. Math. Phys., 47 (2006), 23 pp. doi: 10.1063/1.2245207.

[30]

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.

[31]

B. GuoD. Huang and W. Wang, Diffusion limit of 3D primitive equations of the large-scale ocean under fast oscillating random force, J. Differential Equations, 259 (2015), 2388-2407.  doi: 10.1016/j.jde.2015.03.041.

[32]

N. Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 17, (2007), 159–179. doi: 10.3934/dcds.2007.17.159.

[33]

R. Klein, Scale-dependent models for atmospheric flows, Annual Review of Fluid Mechanics, 42 (2010), 249-274.  doi: 10.1146/annurev-fluid-121108-145537.

[34]

O. Kreml, Š. Nečasová and T. Piasecki, Local existence of strong solution and weak-strong uniqueness for the compressible Navier-Stokes system on moving domains, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, (2019), 1–46. doi: 10.1017/prm.2018.165.

[35]

I. KukavicaR. TemamV. C. Vicol and M. Ziane, Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain, J. Differential Equations, 250 (2011), 1719-1746.  doi: 10.1016/j.jde.2010.07.032.

[36]

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.

[37]

J. Li and Z. Xin, Global existence of weak solutions to the barotropic compressible Navier-Stokes flows with degenerate viscosities, preprint, arXiv: 1504.06826.

[38]

J.-L. LionsO. P. ManleyR. Temam and S. H. Wang, Physical interpretation of the attractor dimension for the primitive equations of atmospheric circulation, J. Atmospheric Sci., 54 (1997), 1137-1143.  doi: 10.1175/1520-0469(1997)054<1137:PIOTAD>2.0.CO;2.

[39]

J.-L. LionsR. Temam and S. H. 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. LionsR. Temam and S. H. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.  doi: 10.1088/0951-7715/5/2/001.

[41]

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

[42]

X. Liu and E. S. Titi, Local well-posedness of strong solutions to the three-dimensional compressible primitive equations, preprint, arXiv: 1806.09868.

[43]

X. Liu and E. S. Titi, Global existence of weak solutions to the compressible primitive equations of atmosphereic dynamics with degenerate viscositites, SIAM J. Math. Anal., 51 (2019), 1913-1964.  doi: 10.1137/18M1211994.

[44]

X. Liu and E. S. Titi, Zero mach number limit of the compressible primitive equations part Ⅰ: Well-prepared initial data, preprint, arXiv: 1905.09367.

[45]

N. Masmoudi and T. K. Wong, On the $H^s$ theory of hydrostatic Euler equations, Arch. Ration. Mech. Anal., 204 (2012), 231-271.  doi: 10.1007/s00205-011-0485-0.

[46]

T. Şengül and S. Wang, Dynamic transitions and baroclinic instability for 3D continuously stratified Boussinesq flows, J. Math. Fluid Mech., 20 (2018), 1173-1193.  doi: 10.1007/s00021-018-0361-x.

[47]

T. Tang and H. Gao, On the stability of weak solution for compressible primitive equations, Acta Appl. Math., 140 (2015), 133-145.  doi: 10.1007/s10440-014-9982-0.

[48]

R. Temam and M. Ziane, Some Mathematical Problems in Geophysical Fluid Dynamics, Vol. 3, Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, 2004.

[49]

A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974.  doi: 10.1007/s00222-016-0666-4.

[50]

F. Wang, C. Dou and Q. Jiu, Global weak solutions to 3D compressible primitive equations with density-dependent viscosity, arXiv: 1712.04180.

[51]

S. Wang and P. Yang, Remarks on the Rayleigh-Bénard convection on spherical shells, J. Math. Fluid Mech., 15 (2013), 537-552.  doi: 10.1007/s00021-012-0128-8.

show all references

References:
[1]

Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity, 12 (1999), 495-512.  doi: 10.1088/0951-7715/12/3/004.

[2]

Y. Brenier, Remarks on the derivation of the hydrostatic Euler equations, Bull. Sci. Math., 127 (2003), 585-595.  doi: 10.1016/S0007-4497(03)00024-1.

[3]

D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl., 86 (2006), 362-368.  doi: 10.1016/j.matpur.2006.06.005.

[4]

D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., 87 (2007), 57-90.  doi: 10.1016/j.matpur.2006.11.001.

[5]

D. BreschF. Guillén-GonzálezN. Masmoudi and M. A. Rodríguez-Bellido, On the uniqueness of weak solutions of the two-dimensional primitive equations, Differential Integral Equations, 16 (2003), 77-94. 

[6]

D. Bresch and P.-E. Jabin, Global existence of weak solutions for compressible Navier-Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. of Math., 188 (2018), 577-684.  doi: 10.4007/annals.2018.188.2.4.

[7]

D. BreschA. Kazhikhov and J. Lemoine, On the two-dimensional hydrostatic Navier-Stokes equations, SIAM J. Math. Anal., 36 (2004/05), 796-814.  doi: 10.1137/S0036141003422242.

[8]

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.

[9]

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. Ration. Mech. Anal., 214 (2014), 35-76.  doi: 10.1007/s00205-014-0752-y.

[10]

C. CaoJ. Li and E. S. Titi, Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion, Comm. Pure Appl. Math., 69 (2016), 1492-1531.  doi: 10.1002/cpa.21576.

[11]

C. CaoJ. Li and E. S. Titi, Strong solutions to the 3D primitive equations with only horizontal dissipation: Near $H^1$ initial data, J. Funct. Anal., 272 (2017), 4606-4641.  doi: 10.1016/j.jfa.2017.01.018.

[12]

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.

[13] J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Mathematical Geophysics: An Introduction to Rotating Fluids and the Navier-Stokes Equations, The Clarendon Press, Oxford University Press, Oxford, 2006. 
[14]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.

[15]

A. DebusscheN. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.

[16]

A. DebusscheN. Glatt-HoltzR. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118.  doi: 10.1088/0951-7715/25/7/2093.

[17]

Z. DongJ. Zhai and R. Zhang, Large deviation principles for 3D stochastic primitive equations, J. Differential Equations, 263 (2017), 3110-3146.  doi: 10.1016/j.jde.2017.04.025.

[18]

M. Ersoy and T. Ngom, Existence of a global weak solution to compressible primitive equations, C. R. Math. Acad. Sci. Paris, 350 (2012), 379-382.  doi: 10.1016/j.crma.2012.04.013.

[19]

M. ErsoyT. Ngom and M. Sy, Compressible primitive equations: Formal derivation and stability of weak solutions, Nonlinearity, 24 (2011), 79-96.  doi: 10.1088/0951-7715/24/1/004.

[20]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Vol. 26, Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2004.

[21]

E. FeireislI. Gallagher and A. Novotný, A singular limit for compressible rotating fluids, SIAM J. Math. Anal., 44 (2012), 192-205.  doi: 10.1137/100808010.

[22]

E. FeireislB. J. Jin and A. Novotný, Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system, J. Math. Fluid Mech., 14 (2012), 717-730.  doi: 10.1007/s00021-011-0091-9.

[23]

E. FeireislB. J. Jin and A. Novotný, Inviscid incompressible limits of strongly stratified fluids, Asymptot. Anal., 89 (2014), 307-329.  doi: 10.3233/ASY-141231.

[24]

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.

[25]

H. Gao and C. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3053-3073.  doi: 10.3934/dcdsb.2016087.

[26]

B. V. Gatapov and A. V. Kazhikhov, Existence of a global solution of a model problem of atmospheric dynamics, Siberian Math. J., 46 (2005), 805-812.  doi: 10.1007/s11202-005-0079-x.

[27]

P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 13 (2011), 137-146.  doi: 10.1007/s00021-009-0006-1.

[28]

F. Guillén-GonzálezN. Masmoudi and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations, Differential Integral Equations, 14 (2001), 1381-1408. 

[29]

B. Guo and D. Huang, Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics, J. Math. Phys., 47 (2006), 23 pp. doi: 10.1063/1.2245207.

[30]

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.

[31]

B. GuoD. Huang and W. Wang, Diffusion limit of 3D primitive equations of the large-scale ocean under fast oscillating random force, J. Differential Equations, 259 (2015), 2388-2407.  doi: 10.1016/j.jde.2015.03.041.

[32]

N. Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 17, (2007), 159–179. doi: 10.3934/dcds.2007.17.159.

[33]

R. Klein, Scale-dependent models for atmospheric flows, Annual Review of Fluid Mechanics, 42 (2010), 249-274.  doi: 10.1146/annurev-fluid-121108-145537.

[34]

O. Kreml, Š. Nečasová and T. Piasecki, Local existence of strong solution and weak-strong uniqueness for the compressible Navier-Stokes system on moving domains, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, (2019), 1–46. doi: 10.1017/prm.2018.165.

[35]

I. KukavicaR. TemamV. C. Vicol and M. Ziane, Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain, J. Differential Equations, 250 (2011), 1719-1746.  doi: 10.1016/j.jde.2010.07.032.

[36]

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.

[37]

J. Li and Z. Xin, Global existence of weak solutions to the barotropic compressible Navier-Stokes flows with degenerate viscosities, preprint, arXiv: 1504.06826.

[38]

J.-L. LionsO. P. ManleyR. Temam and S. H. Wang, Physical interpretation of the attractor dimension for the primitive equations of atmospheric circulation, J. Atmospheric Sci., 54 (1997), 1137-1143.  doi: 10.1175/1520-0469(1997)054<1137:PIOTAD>2.0.CO;2.

[39]

J.-L. LionsR. Temam and S. H. 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. LionsR. Temam and S. H. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.  doi: 10.1088/0951-7715/5/2/001.

[41]

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

[42]

X. Liu and E. S. Titi, Local well-posedness of strong solutions to the three-dimensional compressible primitive equations, preprint, arXiv: 1806.09868.

[43]

X. Liu and E. S. Titi, Global existence of weak solutions to the compressible primitive equations of atmosphereic dynamics with degenerate viscositites, SIAM J. Math. Anal., 51 (2019), 1913-1964.  doi: 10.1137/18M1211994.

[44]

X. Liu and E. S. Titi, Zero mach number limit of the compressible primitive equations part Ⅰ: Well-prepared initial data, preprint, arXiv: 1905.09367.

[45]

N. Masmoudi and T. K. Wong, On the $H^s$ theory of hydrostatic Euler equations, Arch. Ration. Mech. Anal., 204 (2012), 231-271.  doi: 10.1007/s00205-011-0485-0.

[46]

T. Şengül and S. Wang, Dynamic transitions and baroclinic instability for 3D continuously stratified Boussinesq flows, J. Math. Fluid Mech., 20 (2018), 1173-1193.  doi: 10.1007/s00021-018-0361-x.

[47]

T. Tang and H. Gao, On the stability of weak solution for compressible primitive equations, Acta Appl. Math., 140 (2015), 133-145.  doi: 10.1007/s10440-014-9982-0.

[48]

R. Temam and M. Ziane, Some Mathematical Problems in Geophysical Fluid Dynamics, Vol. 3, Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, 2004.

[49]

A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974.  doi: 10.1007/s00222-016-0666-4.

[50]

F. Wang, C. Dou and Q. Jiu, Global weak solutions to 3D compressible primitive equations with density-dependent viscosity, arXiv: 1712.04180.

[51]

S. Wang and P. Yang, Remarks on the Rayleigh-Bénard convection on spherical shells, J. Math. Fluid Mech., 15 (2013), 537-552.  doi: 10.1007/s00021-012-0128-8.

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