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

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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

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

Abstract
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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 & Applied Analysis, 2014, 13 (1) : 135-155. doi: 10.3934/cpaa.2014.13.135

References:

[1]	E. Becker, "Gasdynamik,", Teubner-Verlag, (1966). Google Scholar
[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. doi: 10.1007/s00220-006-0052-y. Google Scholar
[3]	S. Eliezer, A. Ghatak and H. Hora, "An Introduction to Equations of States, Theory and Applications,", Cambridge University Press, (1986). Google Scholar
[4]	E. Feireisl, Incompressible limits and propagation of acoustic waves in large domains with boundaries,, Comm. Math. Phys., 294 (2010), 73. doi: 10.1007/s00220-009-0954-6. Google Scholar
[5]	E. Feireisl, Stability of flows of real monoatomic gases,, Commun. Partial Differential Equations, 31 (2006), 325. doi: 10.1080/03605300500358186. Google Scholar
[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. doi: 10.1007/s00205-007-0066-4. Google Scholar
[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. Google Scholar
[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. doi: 10.1080/03605300903279377. Google Scholar
[9]	X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows,, Comm. Math. Phys., 283 (2008), 255. doi: 10.1007/s00220-008-0497-2. Google Scholar
[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. doi: 10.1002/cpa.3160340405. Google Scholar
[11]	Y.-S. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows,, J. Differential Equations., 251 (2011), 1990. doi: 10.1016/j.jde.2011.04.016. Google Scholar
[12]	Peter Kukucka, Singular Limits of the Equations of Magnetohydrodynamics,, J. Math. Fluid Mech., 13 (2011), 173. doi: 10.1007/s00021-009-0007-0. Google Scholar
[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. doi: 10.1016/j.jmaa.2011.08.070. Google Scholar
[14]	P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid,, J. Math. Pures Appl., 77 (1998), 585. doi: 10.1016/S0021-7824(98)80139-6. Google Scholar
[15]	A. Novotný, M. Ruzicka and G. Thater, Rigorous derivation of the anelastic approximation to the Oberbeck-Boussinesq equations,, Asymptot. Anal., 75 (2011), 93. Google Scholar
[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. doi: 10.1142/S0218202511005003. Google Scholar
[17]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics.IV. Analysis of Operator,", New York: Academy Press [Harcourt Brace Jovanovich Publishers], (1978). Google Scholar

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References:

[1]	E. Becker, "Gasdynamik,", Teubner-Verlag, (1966). Google Scholar
[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. doi: 10.1007/s00220-006-0052-y. Google Scholar
[3]	S. Eliezer, A. Ghatak and H. Hora, "An Introduction to Equations of States, Theory and Applications,", Cambridge University Press, (1986). Google Scholar
[4]	E. Feireisl, Incompressible limits and propagation of acoustic waves in large domains with boundaries,, Comm. Math. Phys., 294 (2010), 73. doi: 10.1007/s00220-009-0954-6. Google Scholar
[5]	E. Feireisl, Stability of flows of real monoatomic gases,, Commun. Partial Differential Equations, 31 (2006), 325. doi: 10.1080/03605300500358186. Google Scholar
[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. doi: 10.1007/s00205-007-0066-4. Google Scholar
[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. Google Scholar
[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. doi: 10.1080/03605300903279377. Google Scholar
[9]	X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows,, Comm. Math. Phys., 283 (2008), 255. doi: 10.1007/s00220-008-0497-2. Google Scholar
[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. doi: 10.1002/cpa.3160340405. Google Scholar
[11]	Y.-S. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows,, J. Differential Equations., 251 (2011), 1990. doi: 10.1016/j.jde.2011.04.016. Google Scholar
[12]	Peter Kukucka, Singular Limits of the Equations of Magnetohydrodynamics,, J. Math. Fluid Mech., 13 (2011), 173. doi: 10.1007/s00021-009-0007-0. Google Scholar
[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. doi: 10.1016/j.jmaa.2011.08.070. Google Scholar
[14]	P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid,, J. Math. Pures Appl., 77 (1998), 585. doi: 10.1016/S0021-7824(98)80139-6. Google Scholar
[15]	A. Novotný, M. Ruzicka and G. Thater, Rigorous derivation of the anelastic approximation to the Oberbeck-Boussinesq equations,, Asymptot. Anal., 75 (2011), 93. Google Scholar
[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. doi: 10.1142/S0218202511005003. Google Scholar
[17]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics.IV. Analysis of Operator,", New York: Academy Press [Harcourt Brace Jovanovich Publishers], (1978). Google Scholar

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