# American Institute of Mathematical Sciences

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

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.
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

show all references

##### 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
 [1] Colette Guillopé, Zaynab Salloum, Raafat Talhouk. Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1001-1028. doi: 10.3934/dcdsb.2010.14.1001 [2] Bernard Ducomet, Šárka Nečasová. On the motion of rigid bodies in an incompressible or compressible viscous fluid under the action of gravitational forces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1193-1213. doi: 10.3934/dcdss.2013.6.1193 [3] Bo-Qing Dong, Zhi-Min Chen. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 765-784. doi: 10.3934/dcds.2009.23.765 [4] Tong Tang, Hongjun Gao. Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1617-1633. doi: 10.3934/dcdsb.2016014 [5] Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161 [6] Fei Jiang, Song Jiang, Weiwei Wang. Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1853-1898. doi: 10.3934/dcdss.2016076 [7] Cristian A. Coclici, Jörg Heiermann, Gh. Moroşanu, W. L. Wendland. Asymptotic analysis of a two--dimensional coupled problem for compressible viscous flows. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 137-163. doi: 10.3934/dcds.2004.10.137 [8] Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503 [9] Takayuki Kubo, Yoshihiro Shibata, Kohei Soga. On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3741-3774. doi: 10.3934/dcds.2016.36.3741 [10] Eduard Feireisl, Hana Petzeltová. Low Mach number asymptotics for reacting compressible fluid flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 455-480. doi: 10.3934/dcds.2010.26.455 [11] Quan Wang. Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 543-563. doi: 10.3934/dcdsb.2014.19.543 [12] Šárka Nečasová, Joerg Wolf. On the existence of global strong solutions to the equations modeling a motion of a rigid body around a viscous fluid. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1539-1562. doi: 10.3934/dcds.2016.36.1539 [13] Eduard Feireisl, Dalibor Pražák. A stabilizing effect of a high-frequency driving force on the motion of a viscous, compressible, and heat conducting fluid. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 95-111. doi: 10.3934/dcdss.2009.2.95 [14] Paul Deuring, Stanislav Kračmar, Šárka Nečasová. A representation formula for linearized stationary incompressible viscous flows around rotating and translating bodies. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 237-253. doi: 10.3934/dcdss.2010.3.237 [15] Xin Liu. Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities. Communications on Pure & Applied Analysis, 2019, 18 (2) : 751-794. doi: 10.3934/cpaa.2019037 [16] Muhammad Bilal Riaz, Naseer Ahmad Asif, Abdon Atangana, Muhammad Imran Asjad. Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 645-664. doi: 10.3934/dcdss.2019041 [17] Xianpeng Hu, Hao Wu. Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3437-3461. doi: 10.3934/dcds.2015.35.3437 [18] Haiyan Yin. The stability of contact discontinuity for compressible planar magnetohydrodynamics. Kinetic & Related Models, 2017, 10 (4) : 1235-1253. doi: 10.3934/krm.2017047 [19] Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124 [20] Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673

2018 Impact Factor: 0.925