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April 2018, 38(4): 1669-1705. doi: 10.3934/dcds.2018069

Low Mach number limit for the compressible magnetohydrodynamic equations in a periodic domain

1. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

2. 

School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210023, China

* Corresponding author: Yanmin Mu.

Received  January 2016 Revised  October 2017 Published  January 2018

This paper studies the convergence of the compressible isentropic magnetohydrodynamic equations to the corresponding incompressiblemagnetohydrodynamic equations with ill-preparedinitial data in a periodic domain.We prove that the solution to the compressible isentropic magnetohydrodynamic equations with small Mach number exists uniformly in the time interval as long as that to the incompressible one does. Furthermore,we obtain the convergence result for the solutions filtered by the group of acoustics.

Citation: Fucai Li, Yanmin Mu. Low Mach number limit for the compressible magnetohydrodynamic equations in a periodic domain. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1669-1705. doi: 10.3934/dcds.2018069
References:
[1]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246. doi: 10.24033/asens.1404.

[2]

F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 198 (2010), 233-271. doi: 10.1007/s00205-010-0306-x.

[3]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131.

[4]

Q.-L. ChenC.-X. Miao and Z.-F. Zhang, Global well-posedness for compressible navier-stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224. doi: 10.1002/cpa.20325.

[5]

R. Danchin, Global existence in critical spaces for compressible navier-stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.

[6]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233. doi: 10.1081/PDE-100106132.

[7]

R. Danchin, On the uniqueness in critical spaces for compressible navier-stokes equations, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 111-128. doi: 10.1007/s00030-004-2032-2.

[8]

R. Danchin, Well-posedness in critical spaces for barotropic viscous fuids with truly not constant density, Comm. Partial Differential Equations, 32 (2007), 1373-1397. doi: 10.1080/03605300600910399.

[9]

R. Danchin, Zero Mach number limit in critial spaces for compressible navier-stokes equations, Ann. Sci. Éc. Norm. Supér.(4), 35 (2002), 27-75. doi: 10.1016/S0012-9593(01)01085-0.

[10]

R. Danchin, Zero Mach number limit for compressible flows with periodic boundary conditions, Amer. J. Math., 124 (2002), 1153-1219. doi: 10.1353/ajm.2002.0036.

[11]

C.-S. DouS. 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.

[12]

J.-S. FanF.-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.

[13]

E. FeireislA. Novotny and Y. Sun, Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains, Discrete Contin. Dyn. Syst., 34 (2014), 121-143. doi: 10.3934/dcds.2014.34.121.

[14]

I. Gallagher, Applications of Schochet's methods to parabolic equations, J. Math. Pures Appl., 77 (1998), 989-1054. doi: 10.1016/S0021-7824(99)80002-6.

[15]

B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202 (2011), 427-460. doi: 10.1007/s00205-011-0430-2.

[16]

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

[17]

S. JiangQ.-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.

[18]

S. JiangQ.-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.

[19]

S. JiangQ.-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.

[20]

S. Jiang and F.-C. Li, Rigorous derivation of the compressible magnetohydrodynamic equations from the electromagnetic fluid system, Nonlinearity, 25 (2012), 1735-1752. doi: 10.1088/0951-7715/25/6/1735.

[21]

F.-C. LiY.-M. Mu and D.-H. Wang, Local well-posedness and low mach number limit of the compressible magnetohydrodynamic equations in critical spaces, Kinetic and Related 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]

N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system, Ann. Inst. H. Poincaŕe Anal. Non Linéaire, 18 (2001), 199-224. doi: 10.1016/S0294-1449(00)00123-2.

[24]

Y.-M. Mu, Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces, Kinet. Relat. Models, 7 (2014), 739-753. doi: 10.3934/krm.2014.7.739.

[25]

S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512. doi: 10.1006/jdeq.1994.1157.

show all references

References:
[1]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246. doi: 10.24033/asens.1404.

[2]

F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 198 (2010), 233-271. doi: 10.1007/s00205-010-0306-x.

[3]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131.

[4]

Q.-L. ChenC.-X. Miao and Z.-F. Zhang, Global well-posedness for compressible navier-stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224. doi: 10.1002/cpa.20325.

[5]

R. Danchin, Global existence in critical spaces for compressible navier-stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.

[6]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233. doi: 10.1081/PDE-100106132.

[7]

R. Danchin, On the uniqueness in critical spaces for compressible navier-stokes equations, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 111-128. doi: 10.1007/s00030-004-2032-2.

[8]

R. Danchin, Well-posedness in critical spaces for barotropic viscous fuids with truly not constant density, Comm. Partial Differential Equations, 32 (2007), 1373-1397. doi: 10.1080/03605300600910399.

[9]

R. Danchin, Zero Mach number limit in critial spaces for compressible navier-stokes equations, Ann. Sci. Éc. Norm. Supér.(4), 35 (2002), 27-75. doi: 10.1016/S0012-9593(01)01085-0.

[10]

R. Danchin, Zero Mach number limit for compressible flows with periodic boundary conditions, Amer. J. Math., 124 (2002), 1153-1219. doi: 10.1353/ajm.2002.0036.

[11]

C.-S. DouS. 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.

[12]

J.-S. FanF.-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.

[13]

E. FeireislA. Novotny and Y. Sun, Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains, Discrete Contin. Dyn. Syst., 34 (2014), 121-143. doi: 10.3934/dcds.2014.34.121.

[14]

I. Gallagher, Applications of Schochet's methods to parabolic equations, J. Math. Pures Appl., 77 (1998), 989-1054. doi: 10.1016/S0021-7824(99)80002-6.

[15]

B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202 (2011), 427-460. doi: 10.1007/s00205-011-0430-2.

[16]

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

[17]

S. JiangQ.-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.

[18]

S. JiangQ.-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.

[19]

S. JiangQ.-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.

[20]

S. Jiang and F.-C. Li, Rigorous derivation of the compressible magnetohydrodynamic equations from the electromagnetic fluid system, Nonlinearity, 25 (2012), 1735-1752. doi: 10.1088/0951-7715/25/6/1735.

[21]

F.-C. LiY.-M. Mu and D.-H. Wang, Local well-posedness and low mach number limit of the compressible magnetohydrodynamic equations in critical spaces, Kinetic and Related 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]

N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system, Ann. Inst. H. Poincaŕe Anal. Non Linéaire, 18 (2001), 199-224. doi: 10.1016/S0294-1449(00)00123-2.

[24]

Y.-M. Mu, Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces, Kinet. Relat. Models, 7 (2014), 739-753. doi: 10.3934/krm.2014.7.739.

[25]

S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512. doi: 10.1006/jdeq.1994.1157.

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