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October  2019, 24(10): 5503-5522. doi: 10.3934/dcdsb.2019068

Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition

Lan Zeng 1,, , Guoxi Ni 2, and  Yingying Li 2,

1. 

School of Mathematical Sciences, Peking University, Beijing 100871, China
2. 

Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

* Corresponding author: Lan Zeng

Received  June 2018 Revised  October 2018 Published  October 2019 Early access  April 2019

Fund Project: The second author is supported by Science Challenge Project, No.TZ2016002.

In this paper, we consider the low Mach number limit of the full compressible MHD equations in a 3-D bounded domain with Dirichlet boundary condition for velocity field, Neumann boundary condition for temperature and perfectly conducting boundary condition for magnetic field. First, the uniform estimates in the Mach number for the strong solutions are obtained in a short time interval, provided that the initial density and temperature are close to the constant states. Then, we prove the solutions of the full compressible MHD equations converge to the isentropic incompressible MHD equations as the Mach number tends to zero.

Keywords: Incompressible limit, full MHD equations, Dirichlet boundary condition.
Mathematics Subject Classification: Primary: 76W05, 76M45; Secondary: 35B40.
Citation: Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5503-5522. doi: 10.3934/dcdsb.2019068
References:
[1]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.

[2]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Functional Analysis, 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.

[3]

W. Q. Cui, Y. B. Ou and D. D. Ren, Incompressible limit of full compressible magnetohydrodynamic equations with well-prepared data in 3-D bounded domains, J. Math. Anal. Appl., 427 (2015), 263-288.  doi: 10.1016/j.jmaa.2015.02.049.

[4]

B. Desjardins, E. Grenier, P. L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl., 78 (1999), 461-471.  doi: 10.1016/S0021-7824(99)00032-X.

[5]

C. S. Dou, S. Jiang and Q. C. Ju, Global existence and the low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain with perfectly conducting boundary, Z. Angew. Math. Phys., 64 (2013), 1661-1678.  doi: 10.1007/s00033-013-0311-7.

[6]

C. S. Dou, S. Jiang and Y. B. Ou, Low Mach number limit of full Navier-Stokes equations in a 3D bounded domain, J. Differential Equations, 258 (2015), 379-398.  doi: 10.1016/j.jde.2014.09.017.

[7]

C. S. Dou and Q. C. Ju, Low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain for all time, Commun. Math. Sci., 12 (2014), 661-679.  doi: 10.4310/CMS.2014.v12.n4.a3.

[8]

J. S. Fan, F. C. Li and G. Nakamura, Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 2015,387–394. doi: 10.3934/proc.2015.0387.

[9]

J. S. Fan, H. J. Gao and B. L. Guo, Low Mach number limit of the compressible magnetohydrodynamic equations with zero thermal conductivity coefficient, Math. Methods Appl. Sci., 34 (2011), 2181-2188.  doi: 10.1002/mma.1515.

[10]

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

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

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A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.

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G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. Ⅰ. Linearized Steady Problems, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[14]

X. P. Hu and D. H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAMJ. Math. Anal., 41 (2009), 1272-1294.  doi: 10.1137/080723983.

[15]

S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal., 42 (2010), 2539-2553.  doi: 10.1137/100785168.

[16]

S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys., 297 (2010), 371-400.  doi: 10.1007/s00220-010-0992-0.

[17]

S. Jiang, Q. C. Ju and F. C. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365.  doi: 10.1088/0951-7715/25/5/1351.

[18]

S. Jiang, Q. C. Ju, F. C. Li and Z. P. Xing, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.

[19]

S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the nonisentropic ideal magnetohydrodynamic equations, SIAM J. Math. Anal., 48 (2016), 302-319.  doi: 10.1137/15M102842X.

[20]

S. Jiang and Y. B. Ou, Incompressible limit of the non-isentropic Navier-Stokes equations with well-prepared initial data in three-dimensional bounded domains, J. Math. Pures Appl., 96 (2011), 1-28.  doi: 10.1016/j.matpur.2011.01.004.

[21]

F. C. Li, Y. M. Mu and D. H. Wang, Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces, Kinet. Relat. Models, 10 (2017), 741-784.  doi: 10.3934/krm.2017030.

[22]

Y. P. Li, Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations, J. Differential Equations, 252 (2012), 2725-2738.  doi: 10.1016/j.jde.2011.10.002.

[23]

Y. P. Li and W. A. Yong, The Zero Mach number limit of the three-dimensional compressible viscous magnetohydrodynamic equations, Chin. Ann. Math. Ser. B, 36 (2015), 1043-1054.  doi: 10.1007/s11401-015-0918-4.

[24]

J. G. Liu and R. Pego, Stable discretization of magnetohydrodynamics in bounded domains, Commun. Math. Sci., 8 (2010), 234-251.  doi: 10.4310/CMS.2010.v8.n1.a12.

[25]

G. Métivier and S. Schchet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.

[26]

Y. B. Ou, Low Mach number limit of viscous polytropic fluid flows, J. Differential Equations, 251 (2011), 2037-2065.  doi: 10.1016/j.jde.2011.07.009.

[27]

D. D. Ren and Y. B. Ou, Incompressible limit of all-time solutions to 3-D full Navier-Stokes equations for perfect gas with well-prepared initial condition, Z. Angew. Math. Phys., 67 (2016), Art. 103, 27 pp. doi: 10.1007/s00033-016-0698-z.

[28]

W. Rusin, On the inviscid limit for the solutions of two-dimensional incompressible Navier-Stokes equations with slip-type boundary conditions, Nonlinearity, 19 (2006), 1349-1363.  doi: 10.1088/0951-7715/19/6/007.

[29]

A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 607-647. 

[30]

A. Valli and W. M. Zajaczkowski, Navier-stokes for compressible fluid: Global existence and qualitative properties of the solutions in the general case, Commun. Math. Phys., 103 (1986), 259-296.  doi: 10.1007/BF01206939.

[31]

S. Wang and Z. L. Xu, Low Mach number limit of non-isentropic magnetohydrodynamic equations in a bounded domain, Nonlinear Anal., 105 (2014), 102-119.  doi: 10.1016/j.na.2014.01.008.

[32]

Y. L. Xiao and Z. P. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055.  doi: 10.1002/cpa.20187.

show all references

References:
[1]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.

[2]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Functional Analysis, 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.

[3]

W. Q. Cui, Y. B. Ou and D. D. Ren, Incompressible limit of full compressible magnetohydrodynamic equations with well-prepared data in 3-D bounded domains, J. Math. Anal. Appl., 427 (2015), 263-288.  doi: 10.1016/j.jmaa.2015.02.049.

[4]

B. Desjardins, E. Grenier, P. L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl., 78 (1999), 461-471.  doi: 10.1016/S0021-7824(99)00032-X.

[5]

C. S. Dou, S. Jiang and Q. C. Ju, Global existence and the low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain with perfectly conducting boundary, Z. Angew. Math. Phys., 64 (2013), 1661-1678.  doi: 10.1007/s00033-013-0311-7.

[6]

C. S. Dou, S. Jiang and Y. B. Ou, Low Mach number limit of full Navier-Stokes equations in a 3D bounded domain, J. Differential Equations, 258 (2015), 379-398.  doi: 10.1016/j.jde.2014.09.017.

[7]

C. S. Dou and Q. C. Ju, Low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain for all time, Commun. Math. Sci., 12 (2014), 661-679.  doi: 10.4310/CMS.2014.v12.n4.a3.

[8]

J. S. Fan, F. C. Li and G. Nakamura, Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 2015,387–394. doi: 10.3934/proc.2015.0387.

[9]

J. S. Fan, H. J. Gao and B. L. Guo, Low Mach number limit of the compressible magnetohydrodynamic equations with zero thermal conductivity coefficient, Math. Methods Appl. Sci., 34 (2011), 2181-2188.  doi: 10.1002/mma.1515.

[10]

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

[11]

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

[12]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.

[13]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. Ⅰ. Linearized Steady Problems, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[14]

X. P. Hu and D. H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAMJ. Math. Anal., 41 (2009), 1272-1294.  doi: 10.1137/080723983.

[15]

S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal., 42 (2010), 2539-2553.  doi: 10.1137/100785168.

[16]

S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys., 297 (2010), 371-400.  doi: 10.1007/s00220-010-0992-0.

[17]

S. Jiang, Q. C. Ju and F. C. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365.  doi: 10.1088/0951-7715/25/5/1351.

[18]

S. Jiang, Q. C. Ju, F. C. Li and Z. P. Xing, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.

[19]

S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the nonisentropic ideal magnetohydrodynamic equations, SIAM J. Math. Anal., 48 (2016), 302-319.  doi: 10.1137/15M102842X.

[20]

S. Jiang and Y. B. Ou, Incompressible limit of the non-isentropic Navier-Stokes equations with well-prepared initial data in three-dimensional bounded domains, J. Math. Pures Appl., 96 (2011), 1-28.  doi: 10.1016/j.matpur.2011.01.004.

[21]

F. C. Li, Y. M. Mu and D. H. Wang, Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces, Kinet. Relat. Models, 10 (2017), 741-784.  doi: 10.3934/krm.2017030.

[22]

Y. P. Li, Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations, J. Differential Equations, 252 (2012), 2725-2738.  doi: 10.1016/j.jde.2011.10.002.

[23]

Y. P. Li and W. A. Yong, The Zero Mach number limit of the three-dimensional compressible viscous magnetohydrodynamic equations, Chin. Ann. Math. Ser. B, 36 (2015), 1043-1054.  doi: 10.1007/s11401-015-0918-4.

[24]

J. G. Liu and R. Pego, Stable discretization of magnetohydrodynamics in bounded domains, Commun. Math. Sci., 8 (2010), 234-251.  doi: 10.4310/CMS.2010.v8.n1.a12.

[25]

G. Métivier and S. Schchet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.

[26]

Y. B. Ou, Low Mach number limit of viscous polytropic fluid flows, J. Differential Equations, 251 (2011), 2037-2065.  doi: 10.1016/j.jde.2011.07.009.

[27]

D. D. Ren and Y. B. Ou, Incompressible limit of all-time solutions to 3-D full Navier-Stokes equations for perfect gas with well-prepared initial condition, Z. Angew. Math. Phys., 67 (2016), Art. 103, 27 pp. doi: 10.1007/s00033-016-0698-z.

[28]

W. Rusin, On the inviscid limit for the solutions of two-dimensional incompressible Navier-Stokes equations with slip-type boundary conditions, Nonlinearity, 19 (2006), 1349-1363.  doi: 10.1088/0951-7715/19/6/007.

[29]

A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 607-647. 

[30]

A. Valli and W. M. Zajaczkowski, Navier-stokes for compressible fluid: Global existence and qualitative properties of the solutions in the general case, Commun. Math. Phys., 103 (1986), 259-296.  doi: 10.1007/BF01206939.

[31]

S. Wang and Z. L. Xu, Low Mach number limit of non-isentropic magnetohydrodynamic equations in a bounded domain, Nonlinear Anal., 105 (2014), 102-119.  doi: 10.1016/j.na.2014.01.008.

[32]

Y. L. Xiao and Z. P. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055.  doi: 10.1002/cpa.20187.

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