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January  2014, 13(1): 135-155. doi: 10.3934/cpaa.2014.13.135

Incompressible limit for the full magnetohydrodynamics flows under Strong Stratification on unbounded domains

Gyungsoo Woo 1, and  Young-Sam Kwon 2,

1. 

Department of Applied Mathematics, Changwon National University, Changwon 641-773, South Korea
2. 

Department of Mathematics Dong-A University, Busan 604-714

Received  May 2012 Revised  May 2013 Published  July 2013

In this paper we consider the magnetohydrodynamics flows giving rise to a variety of mathematical problems in many areas. We study the incompressible limit problems for magnetohydrodynamics flows under strong stratification on unbounded domains.
Keywords: compressible and viscous fluid, magnetohydrodynamics flows., Incompressible limit, strong stratification.
Mathematics Subject Classification: 35B40, 35D05, 35B4.
Citation: Gyungsoo Woo, Young-Sam Kwon. Incompressible limit for the full magnetohydrodynamics flows under Strong Stratification on unbounded domains. Communications on Pure and Applied Analysis, 2014, 13 (1) : 135-155. doi: 10.3934/cpaa.2014.13.135
References:
[1]

E. Becker, "Gasdynamik," Teubner-Verlag, Stuttgart, 1966.

[2]

B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-625. doi: 10.1007/s00220-006-0052-y.

[3]

S. Eliezer, A. Ghatak and H. Hora, "An Introduction to Equations of States, Theory and Applications," Cambridge University Press, Cambridge, 1986.

[4]

E. Feireisl, Incompressible limits and propagation of acoustic waves in large domains with boundaries, Comm. Math. Phys., 294 (2010), 73-95. doi: 10.1007/s00220-009-0954-6.

[5]

E. Feireisl, Stability of flows of real monoatomic gases, Commun. Partial Differential Equations, 31 (2006), 325-348. doi: 10.1080/03605300500358186.

[6]

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

[7]

E. Feireisl and A. Novotný, "Singular Limit in the Thermodynamics of Viscous Fluids," Advanceds in Mathematical Fluid Mechanics, 2009. doi: 10.1007/978-3-7643-8843-0.

[8]

E. Feireisl, A. Novotný} and H. Petzeltová, Low Mach number limt for the Navier-Stokes system on unbounded domains under strong stratification, Comm. P.D.E., 35 (2010), 68-88. doi: 10.1080/03605300903279377.

[9]

X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284. doi: 10.1007/s00220-008-0497-2.

[10]

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.

[11]

Y.-S. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows, J. Differential Equations., 251 (2011), 1990-2023. doi: 10.1016/j.jde.2011.04.016.

[12]

Peter Kukucka, Singular Limits of the Equations of Magnetohydrodynamics, J. Math. Fluid Mech., 13 (2011), 173-189. doi: 10.1007/s00021-009-0007-0.

[13]

G. Lee, P. Kim and Y.-S. Kwon, Incompressible limit for the full magnetohydrodynamics flows under strong stratification, J. Math. Anal. Appl., 387 (2012), 221-240. doi: 10.1016/j.jmaa.2011.08.070.

[14]

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.

[15]

A. Novotný, M. Ruzicka and G. Thater, Rigorous derivation of the anelastic approximation to the Oberbeck-Boussinesq equations, Asymptot. Anal., 75 (2011), 93-123.

[16]

A. Novotný, M. Ruzicka and G. Thater, Singular limit of the equations of magnetohydrodynamics in the presence of strong stratification, Math. Models Methods Appl. Sci., 21 (2011), 115-147. doi: 10.1142/S0218202511005003.

[17]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics.IV. Analysis of Operator," New York: Academy Press [Harcourt Brace Jovanovich Publishers], 1978.

show all references

References:
[1]

E. Becker, "Gasdynamik," Teubner-Verlag, Stuttgart, 1966.

[2]

B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-625. doi: 10.1007/s00220-006-0052-y.

[3]

S. Eliezer, A. Ghatak and H. Hora, "An Introduction to Equations of States, Theory and Applications," Cambridge University Press, Cambridge, 1986.

[4]

E. Feireisl, Incompressible limits and propagation of acoustic waves in large domains with boundaries, Comm. Math. Phys., 294 (2010), 73-95. doi: 10.1007/s00220-009-0954-6.

[5]

E. Feireisl, Stability of flows of real monoatomic gases, Commun. Partial Differential Equations, 31 (2006), 325-348. doi: 10.1080/03605300500358186.

[6]

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

[7]

E. Feireisl and A. Novotný, "Singular Limit in the Thermodynamics of Viscous Fluids," Advanceds in Mathematical Fluid Mechanics, 2009. doi: 10.1007/978-3-7643-8843-0.

[8]

E. Feireisl, A. Novotný} and H. Petzeltová, Low Mach number limt for the Navier-Stokes system on unbounded domains under strong stratification, Comm. P.D.E., 35 (2010), 68-88. doi: 10.1080/03605300903279377.

[9]

X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284. doi: 10.1007/s00220-008-0497-2.

[10]

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.

[11]

Y.-S. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows, J. Differential Equations., 251 (2011), 1990-2023. doi: 10.1016/j.jde.2011.04.016.

[12]

Peter Kukucka, Singular Limits of the Equations of Magnetohydrodynamics, J. Math. Fluid Mech., 13 (2011), 173-189. doi: 10.1007/s00021-009-0007-0.

[13]

G. Lee, P. Kim and Y.-S. Kwon, Incompressible limit for the full magnetohydrodynamics flows under strong stratification, J. Math. Anal. Appl., 387 (2012), 221-240. doi: 10.1016/j.jmaa.2011.08.070.

[14]

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.

[15]

A. Novotný, M. Ruzicka and G. Thater, Rigorous derivation of the anelastic approximation to the Oberbeck-Boussinesq equations, Asymptot. Anal., 75 (2011), 93-123.

[16]

A. Novotný, M. Ruzicka and G. Thater, Singular limit of the equations of magnetohydrodynamics in the presence of strong stratification, Math. Models Methods Appl. Sci., 21 (2011), 115-147. doi: 10.1142/S0218202511005003.

[17]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics.IV. Analysis of Operator," New York: Academy Press [Harcourt Brace Jovanovich Publishers], 1978.

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