January  2020, 40(1): 395-421. doi: 10.3934/dcds.2020015

Derivation of geostrophic equations as a rigorous limit of compressible rotating and heat conducting fluids with the general initial data

1. 

Department of Mathematics, Dong-A University, Busan, Korea

2. 

University of Toulon, IMATH, BP 20139,839 57 La Garde, France

* Corresponding author: Anotnin Novotny

Received  February 2019 Revised  July 2019 Published  October 2019

Fund Project: The work of Young-Sam Kwon was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF- 2017R1D1A1B03030249)

We investigate a distinguished low Mach and Rossby - high Reynolds and Péclet number singular limit in the complete Navier-Stokes-Fourier system towards a strong solution of a geostrophic system of equations. The limit is effectuated in the context of weak solutions with ill prepared initial data. The main tool in the proof is based on the relative energy method.

Citation: Young-Sam Kwon, Antonin Novotny. Derivation of geostrophic equations as a rigorous limit of compressible rotating and heat conducting fluids with the general initial data. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 395-421. doi: 10.3934/dcds.2020015
References:
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A. BabinA. Mahalov and B. Nicolaeko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana Univ. Math. J., 48 (1999), 1133-1176.   Google Scholar

[2]

A. BabinA. Mahalov and B. Nicolaeko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J., 50 (2001), 1-35.  doi: 10.1512/iumj.2001.50.2155.  Google Scholar

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D. BreschB. Desjardins and D. Gerard-Varet, Rotating fluids in a cylinder, Disc. Cont. Dyn. Syst., 11 (2004), 47-82.  doi: 10.3934/dcds.2004.11.47.  Google Scholar

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T. ClopeauA. Mikelic and R. Robert, On the vanishing viscosity limit for the 2D incomressible Navier-Stokes equations with the friction boundary conditions, Nonlinearity, 11 (1998), 1625-1636.  doi: 10.1088/0951-7715/11/6/011.  Google Scholar

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R. Danchin, Low Mach number for viscous compressible flows, M2AN Math. Model Numer. Anal., 39 (2005), 459-475.  doi: 10.1051/m2an:2005019.  Google Scholar

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D. R. Durran, Is the Coriolis force really responsible for the inertial oscillation?, Bull. Amer. Meteorological Soc., 74 (1993), 2179-2184.  doi: 10.1175/1520-0477(1993)074<2179:ITCFRR>2.0.CO;2.  Google Scholar

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D. B. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraing force, Ann. Math., 105 (1977), 141-200.  doi: 10.2307/1971029.  Google Scholar

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L. C. Evans, Partial Differential Equations, Second 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, Low Mach number limits of compressible rotating fluids, J. Math. Fluid Mechanics, 14 (2012), 61-78.  doi: 10.1007/s00021-010-0043-9.  Google Scholar

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E. FeireislY. Lu and A. Novotny, Rotating compressible fluids under strong stratification, Nonlinear Analysis: Real World Applications, 19 (2014), 11-18.  doi: 10.1016/j.nonrwa.2014.02.004.  Google Scholar

[12]

E. FeireislI. GallagherD. Gerard-Varet and A. Novotný, Multi-scale Analysis of Compressible Viscous and Rotating Fluids, Commun. Math. Phys., 314 (2012), 641-670.  doi: 10.1007/s00220-012-1533-9.  Google Scholar

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E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 204 (2012), 683-706.  doi: 10.1007/s00205-011-0490-3.  Google Scholar

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E. Feireisl and A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system, Commun. Math. Phys., 321 (2013), 605-628.  doi: 10.1007/s00220-013-1691-4.  Google Scholar

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E. Feireisl and A. Novotný, Scale interactions in compressible rotating fluids, Annali di Matematica Pura ed Applicata, 193 (2014), 1703-1725.  doi: 10.1007/s10231-013-0353-7.  Google Scholar

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[21]

E. FeireislY. Lu and A. Novotny, Rotating compressible fluids under strong stratification, Nonlinear Analysis: Real World Applications, 19 (2014), 11-18.  doi: 10.1016/j.nonrwa.2014.02.004.  Google Scholar

[22]

F. Fanelli, Highly rotating viscous compressible fluids in presence of capillarity effects, Math. Ann., 366 (2016), 981-1033.  doi: 10.1007/s00208-015-1358-x.  Google Scholar

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I. Gallagher and L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations, J. Anal. Math., 99 (2006), 1-34.  doi: 10.1007/BF02789441.  Google Scholar

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G.-M. Gie and J. P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Diff. Equations, 235 (2012), 1862-1892.  doi: 10.1016/j.jde.2012.06.008.  Google Scholar

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D. Iftimie and F. Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip condidions, Arch. Rat. Mech. Anal., 199 (2011), 145-175.  doi: 10.1007/s00205-010-0320-z.  Google Scholar

[30]

D. JessléB. J. Jin and A. Novotný, Navier-Stokes-Fourier system on unbounded domains: Weak solutions, relative entropies, weak-strong uniqueness, SIAM J. Math. Anal., 45 (2013), 1907-1951.  doi: 10.1137/120874576.  Google Scholar

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[32]

T. Kato and G. Ponce, Well-Posedness of the Euler and Navier-Stokes Equations in the Lebesgue Spaces $L^p_ s(R2)$, Revista Matematica Iberoamericana, 2 (1986), 73-88.  doi: 10.4171/RMI/26.  Google Scholar

[33]

T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct.Anal., 56 (1984), 15-28.  doi: 10.1016/0022-1236(84)90024-7.  Google Scholar

[34]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.  Google Scholar

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R. Klein, Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods, Z. Angw. Math. Mech., 80 (2000), 765-777.  doi: 10.1002/1521-4001(200011)80:11/12<765::AID-ZAMM765>3.0.CO;2-1.  Google Scholar

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Y. KwonD. Maltese and A. Novotny, Multiscale analysis in the Compressible Rotating and Heat Conducting Fluid, J. Math. Fluid. Mech., 20 (2018), 421-444.  doi: 10.1007/s00021-017-0327-4.  Google Scholar

[38]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J.Math.Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6.  Google Scholar

[39]

N. Masmoudi and F. Rousset, Uniform regularity for the Navier-Stokes equations with Navier boundary conditions, Arch. Rational Mech. Anal., 203 (2012), 529-575.  doi: 10.1007/s00205-011-0456-5.  Google Scholar

[40]

N. Masmoudi, The Euler limit of the Navier-Stokes equations and rotating fluid with boundary, Arch. Rational Mech. Anal., 142 (1998), 375-394.  doi: 10.1007/s002050050097.  Google Scholar

[41]

N. Masmoudi, Incompressible inviscid limit of the compressible Navier–Stokes system, Ann. Inst. H. Poincaré, Anal. non linéaire, 18 (2001), 199–224. doi: 10.1016/S0294-1449(00)00123-2.  Google Scholar

[42]

N. Masmoudi, Examples of singular limits in hydrodynamics, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 3 (2007), 195–275. doi: 10.1016/S1874-5717(07)80006-5.  Google Scholar

[43]

M. Oliver, Classical sofor a generalized Euler equation in two dimesions, J. Math. Anal. Appl. (9), 215 (1997), 471-484.  doi: 10.1006/jmaa.1997.5647.  Google Scholar

[44]

M. Sammartino and R. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space Ⅰ, Existence for Euler and Prandtle equations, Comm. Math. Phys., 192 (1998), 433-461.  doi: 10.1007/s002200050304.  Google Scholar

[45]

M. Sammartino and R. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space Ⅱ, Construction of Navier-Stokes equations, Comm. Math. Phys., 192 (1998), 463-491.  doi: 10.1007/s002200050305.  Google Scholar

[46]

S. Schochet, The mathematical theory of low Mach number flows, M2AN Math. Model Numer. anal., 39 (2005), 441-458.  doi: 10.1051/m2an:2005017.  Google Scholar

[47]

H. S. G. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in $R^3$, Trans. Amer.Math. Soc., 157 (1971), 373-397.  doi: 10.2307/1995853.  Google Scholar

[48]

R. Temam and X. Wang, On the behaviour of the solutions of the Navier-Stokes equations at vanishing viscosity, Annani Scuola Normale Pisa, 25 (1997), 807-828.   Google Scholar

[49]

R. Temam and X. Wang, Boundary layer associated with incompressible Navier-Stokes equations: The noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686.  doi: 10.1006/jdeq.2001.4038.  Google Scholar

[50] G. K. Vallis, Atmospheric and Ocean Fluid Dynamics,, Cambridge University Press, Cambridge, 2006.   Google Scholar
[51]

L. WangZ. Xin and A. Zang, Vanishsing viscous limits for 3D Navier-Stokes equations with a Navier-slip boundary condidtions, J. Math. Fluid. Mech., 14 (2012), 791-825.  doi: 10.1007/s00021-012-0103-4.  Google Scholar

show all references

References:
[1]

A. BabinA. Mahalov and B. Nicolaeko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana Univ. Math. J., 48 (1999), 1133-1176.   Google Scholar

[2]

A. BabinA. Mahalov and B. Nicolaeko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J., 50 (2001), 1-35.  doi: 10.1512/iumj.2001.50.2155.  Google Scholar

[3]

D. BreschB. Desjardins and D. Gerard-Varet, Rotating fluids in a cylinder, Disc. Cont. Dyn. Syst., 11 (2004), 47-82.  doi: 10.3934/dcds.2004.11.47.  Google Scholar

[4] J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Mathematical Geophysics, volume 32 of Oxford Lecture Series in Mathematics and Applications, The Clarendon Press Oxford University Press, Oxford, 2006.   Google Scholar
[5]

T. ClopeauA. Mikelic and R. Robert, On the vanishing viscosity limit for the 2D incomressible Navier-Stokes equations with the friction boundary conditions, Nonlinearity, 11 (1998), 1625-1636.  doi: 10.1088/0951-7715/11/6/011.  Google Scholar

[6]

R. Danchin, Low Mach number for viscous compressible flows, M2AN Math. Model Numer. Anal., 39 (2005), 459-475.  doi: 10.1051/m2an:2005019.  Google Scholar

[7]

D. R. Durran, Is the Coriolis force really responsible for the inertial oscillation?, Bull. Amer. Meteorological Soc., 74 (1993), 2179-2184.  doi: 10.1175/1520-0477(1993)074<2179:ITCFRR>2.0.CO;2.  Google Scholar

[8]

D. B. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraing force, Ann. Math., 105 (1977), 141-200.  doi: 10.2307/1971029.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[10]

E. Feireisl, Low Mach number limits of compressible rotating fluids, J. Math. Fluid Mechanics, 14 (2012), 61-78.  doi: 10.1007/s00021-010-0043-9.  Google Scholar

[11]

E. FeireislY. Lu and A. Novotny, Rotating compressible fluids under strong stratification, Nonlinear Analysis: Real World Applications, 19 (2014), 11-18.  doi: 10.1016/j.nonrwa.2014.02.004.  Google Scholar

[12]

E. FeireislI. GallagherD. Gerard-Varet and A. Novotný, Multi-scale Analysis of Compressible Viscous and Rotating Fluids, Commun. Math. Phys., 314 (2012), 641-670.  doi: 10.1007/s00220-012-1533-9.  Google Scholar

[13]

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

[14]

E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 204 (2012), 683-706.  doi: 10.1007/s00205-011-0490-3.  Google Scholar

[15]

E. Feireisl and A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system, Commun. Math. Phys., 321 (2013), 605-628.  doi: 10.1007/s00220-013-1691-4.  Google Scholar

[16]

E. Feireisl and A. Novotný, Inviscid incompressible limits under mild stratification: A rigorous derivation of the Euler-Boussinesq system, Applied Mathematics and Optimization, 70 (2014), 279-307.  doi: 10.1007/s00245-014-9243-7.  Google Scholar

[17]

E. Feireisl and A. Novotný, The low mach number limit for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 186 (2007), 77-107.  doi: 10.1007/s00205-007-0066-4.  Google Scholar

[18]

E. Feireisl and A. Novotný, Multiple scales and singular limits for compressible rotating fluids with general initial data, Comm. Part. Diff. Eq., 39 (2014), 1104-1127.  doi: 10.1080/03605302.2013.856917.  Google Scholar

[19]

E. Feireisl and A. Novotný, Scale interactions in compressible rotating fluids, Annali di Matematica Pura ed Applicata, 193 (2014), 1703-1725.  doi: 10.1007/s10231-013-0353-7.  Google Scholar

[20]

E. FeireislB. J. Jin and A. Novotny, Relative Entropies, Suitable Weak Solutions, and Weak-Strong Uniqueness for the Compressible Navier–Stokes System, Journal of Mathematical Fluid Mechanics, 14 (2012), 717-730.  doi: 10.1007/s00021-011-0091-9.  Google Scholar

[21]

E. FeireislY. Lu and A. Novotny, Rotating compressible fluids under strong stratification, Nonlinear Analysis: Real World Applications, 19 (2014), 11-18.  doi: 10.1016/j.nonrwa.2014.02.004.  Google Scholar

[22]

F. Fanelli, Highly rotating viscous compressible fluids in presence of capillarity effects, Math. Ann., 366 (2016), 981-1033.  doi: 10.1007/s00208-015-1358-x.  Google Scholar

[23]

F. Fanelli, A singular limit problem for rotating capillary fluids with variable rotation axis, J. Math. Fluid Mech., 18 (2016), 625-658.  doi: 10.1007/s00021-016-0256-7.  Google Scholar

[24]

I. Gallagher, Résultats récents sur la limite incompressible, Astérisque, Séminaire Bourbaki, 2003/2004 (2005), 29–57.  Google Scholar

[25]

I. Gallagher and L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations, J. Anal. Math., 99 (2006), 1-34.  doi: 10.1007/BF02789441.  Google Scholar

[26]

I. Gallagher and L. Saint-Raymond, Mathematical study of the betaplane model: Equatorial waves and convergence results, Mem. Soc. Math. Fr., 107 (2006), v+116 pp.  Google Scholar

[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.  Google Scholar

[28]

G.-M. Gie and J. P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Diff. Equations, 235 (2012), 1862-1892.  doi: 10.1016/j.jde.2012.06.008.  Google Scholar

[29]

D. Iftimie and F. Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip condidions, Arch. Rat. Mech. Anal., 199 (2011), 145-175.  doi: 10.1007/s00205-010-0320-z.  Google Scholar

[30]

D. JessléB. J. Jin and A. Novotný, Navier-Stokes-Fourier system on unbounded domains: Weak solutions, relative entropies, weak-strong uniqueness, SIAM J. Math. Anal., 45 (2013), 1907-1951.  doi: 10.1137/120874576.  Google Scholar

[31]

T. Kato, Remarks on the zero viscosity limit for nonstationary Navier–Stokes flows with boundary, Seminar on Nonlinear Partial Differential Equations (Berkeley, Calif., 1983), 85–98, Math. Sci. Res. Inst. Publ., 2, Springer, New York, 1984. doi: 10.1007/978-1-4612-1110-5_6.  Google Scholar

[32]

T. Kato and G. Ponce, Well-Posedness of the Euler and Navier-Stokes Equations in the Lebesgue Spaces $L^p_ s(R2)$, Revista Matematica Iberoamericana, 2 (1986), 73-88.  doi: 10.4171/RMI/26.  Google Scholar

[33]

T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct.Anal., 56 (1984), 15-28.  doi: 10.1016/0022-1236(84)90024-7.  Google Scholar

[34]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.  Google Scholar

[35]

R. Klein, Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods, Z. Angw. Math. Mech., 80 (2000), 765-777.  doi: 10.1002/1521-4001(200011)80:11/12<765::AID-ZAMM765>3.0.CO;2-1.  Google Scholar

[36]

R. Klein, Scale-dependent models for atmospheric flows, Annual Review of Fluid Mechanics. Vol. 42, Annu. Rev. Fluid. Mech., Annual Reviews, Palo Alto, CA, 42 (2010), 249–274. doi: 10.1146/annurev-fluid-121108-145537.  Google Scholar

[37]

Y. KwonD. Maltese and A. Novotny, Multiscale analysis in the Compressible Rotating and Heat Conducting Fluid, J. Math. Fluid. Mech., 20 (2018), 421-444.  doi: 10.1007/s00021-017-0327-4.  Google Scholar

[38]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J.Math.Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6.  Google Scholar

[39]

N. Masmoudi and F. Rousset, Uniform regularity for the Navier-Stokes equations with Navier boundary conditions, Arch. Rational Mech. Anal., 203 (2012), 529-575.  doi: 10.1007/s00205-011-0456-5.  Google Scholar

[40]

N. Masmoudi, The Euler limit of the Navier-Stokes equations and rotating fluid with boundary, Arch. Rational Mech. Anal., 142 (1998), 375-394.  doi: 10.1007/s002050050097.  Google Scholar

[41]

N. Masmoudi, Incompressible inviscid limit of the compressible Navier–Stokes system, Ann. Inst. H. Poincaré, Anal. non linéaire, 18 (2001), 199–224. doi: 10.1016/S0294-1449(00)00123-2.  Google Scholar

[42]

N. Masmoudi, Examples of singular limits in hydrodynamics, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 3 (2007), 195–275. doi: 10.1016/S1874-5717(07)80006-5.  Google Scholar

[43]

M. Oliver, Classical sofor a generalized Euler equation in two dimesions, J. Math. Anal. Appl. (9), 215 (1997), 471-484.  doi: 10.1006/jmaa.1997.5647.  Google Scholar

[44]

M. Sammartino and R. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space Ⅰ, Existence for Euler and Prandtle equations, Comm. Math. Phys., 192 (1998), 433-461.  doi: 10.1007/s002200050304.  Google Scholar

[45]

M. Sammartino and R. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space Ⅱ, Construction of Navier-Stokes equations, Comm. Math. Phys., 192 (1998), 463-491.  doi: 10.1007/s002200050305.  Google Scholar

[46]

S. Schochet, The mathematical theory of low Mach number flows, M2AN Math. Model Numer. anal., 39 (2005), 441-458.  doi: 10.1051/m2an:2005017.  Google Scholar

[47]

H. S. G. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in $R^3$, Trans. Amer.Math. Soc., 157 (1971), 373-397.  doi: 10.2307/1995853.  Google Scholar

[48]

R. Temam and X. Wang, On the behaviour of the solutions of the Navier-Stokes equations at vanishing viscosity, Annani Scuola Normale Pisa, 25 (1997), 807-828.   Google Scholar

[49]

R. Temam and X. Wang, Boundary layer associated with incompressible Navier-Stokes equations: The noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686.  doi: 10.1006/jdeq.2001.4038.  Google Scholar

[50] G. K. Vallis, Atmospheric and Ocean Fluid Dynamics,, Cambridge University Press, Cambridge, 2006.   Google Scholar
[51]

L. WangZ. Xin and A. Zang, Vanishsing viscous limits for 3D Navier-Stokes equations with a Navier-slip boundary condidtions, J. Math. Fluid. Mech., 14 (2012), 791-825.  doi: 10.1007/s00021-012-0103-4.  Google Scholar

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