September  2017, 10(3): 741-784. doi: 10.3934/krm.2017030

Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces

1. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

2. 

School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210046, China

3. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Y. Mu is the corresponding author

Received  July 2016 Revised  October 2016 Published  December 2016

The local well-posedness and low Mach number limit are considered for the multi-dimensional isentropic compressible viscous magnetohydrodynamic equations in critical spaces. First the local well-posedness of solution to the viscous magnetohydrodynamic equations with large initial data is established. Then the low Mach number limit is studied for general large data and it is proved that the solution of the compressible magnetohydrodynamic equations converges to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero. Moreover, the convergence rates are obtained.

Citation: Fucai Li, Yanmin Mu, Dehua Wang. Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2017, 10 (3) : 741-784. doi: 10.3934/krm.2017030
References:
[1]

H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

D. Bian and B. Yuan, Well-posedness in super critical Besov spaces for the compressible MHD equations, Int. J. Dyn. Syst. Differ. Equ., 3 (2011), 383-399. doi: 10.1504/IJDSDE.2011.041882.

[3]

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.

[4]

J. -Y. Chemin, Perfect Incompressible Fluids, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Oxford Lecture Series in Mathematics and its Applications 14, The Clarendon Press, Oxford University Press, New York, 1998.

[5]

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.

[6]

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

[7]

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.

[8]

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.

[9]

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.

[10]

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.

[11]

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.

[12]

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.

[13] E. Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.
[14]

E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.

[15]

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.

[16]

C. C. Hao, Well-posedness to the compressible viscous magnetohydrddynamic system, Nonlinear Anal. Real World Appl., 12 (2011), 2962-2972. doi: 10.1016/j.nonrwa.2011.04.017.

[17]

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.

[18]

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.

[19]

X.-P. Hu and D.-H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9.

[20]

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.

[21]

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.

[22]

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.

[23]

A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, Massachusetts, 1965.

[24]

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1960.

[25]

F.-C. Li and H.-Y. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 109-126. doi: 10.1017/S0308210509001632.

[26]

H.-L. LiX.-Y. Xu and J.-W. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387. doi: 10.1137/120893355.

[27]

X.-L. LiN. Su and D.-H. Wang, Local strong solution to the compressible magnetohydrodynamic flow with large data, J. Hyperbolic Differ. Equ., 8 (2011), 415-436. doi: 10.1142/S0219891611002457.

[28]

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.

[29]

P. -L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford Lecture Series in Mathematics and Its Applications, 10. Oxford Science Publications. Clarendon, Oxford University Press, 1998.

[30]

S. LiuH. Yu and J.-W. Zhang, Global weak solutions of 3D compressible MHD with discontinuous initial data and vacuum, J. Differential Equations, 254 (2013), 229-255. doi: 10.1016/j.jde.2012.08.006.

[31]

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.

[32]

R. V. Polovin and V. P. Demutskii, Fundamentals Of Magnetohydrodynamics, Consultants, Bureau, New York, 1990.

[33]

A. Suen, A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density, Discrete Contin. Dyn. Syst., 33 (2013), 3791-3805. doi: 10.3934/dcds.2013.33.3791.

[34]

A. Suen and D. Hoff, Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 205 (2012), 27-58. doi: 10.1007/s00205-012-0498-3.

[35]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics 78. Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[36]

X. Xu and J. Zhang, A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum, Math. Models Methods Appl. Sci., 22 (2012), 1150010, 23 pp. doi: 10.1142/S0218202511500102.

show all references

References:
[1]

H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

D. Bian and B. Yuan, Well-posedness in super critical Besov spaces for the compressible MHD equations, Int. J. Dyn. Syst. Differ. Equ., 3 (2011), 383-399. doi: 10.1504/IJDSDE.2011.041882.

[3]

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.

[4]

J. -Y. Chemin, Perfect Incompressible Fluids, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Oxford Lecture Series in Mathematics and its Applications 14, The Clarendon Press, Oxford University Press, New York, 1998.

[5]

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.

[6]

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

[7]

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.

[8]

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.

[9]

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.

[10]

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.

[11]

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.

[12]

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.

[13] E. Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.
[14]

E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.

[15]

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.

[16]

C. C. Hao, Well-posedness to the compressible viscous magnetohydrddynamic system, Nonlinear Anal. Real World Appl., 12 (2011), 2962-2972. doi: 10.1016/j.nonrwa.2011.04.017.

[17]

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.

[18]

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.

[19]

X.-P. Hu and D.-H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9.

[20]

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.

[21]

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.

[22]

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.

[23]

A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, Massachusetts, 1965.

[24]

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1960.

[25]

F.-C. Li and H.-Y. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 109-126. doi: 10.1017/S0308210509001632.

[26]

H.-L. LiX.-Y. Xu and J.-W. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387. doi: 10.1137/120893355.

[27]

X.-L. LiN. Su and D.-H. Wang, Local strong solution to the compressible magnetohydrodynamic flow with large data, J. Hyperbolic Differ. Equ., 8 (2011), 415-436. doi: 10.1142/S0219891611002457.

[28]

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.

[29]

P. -L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford Lecture Series in Mathematics and Its Applications, 10. Oxford Science Publications. Clarendon, Oxford University Press, 1998.

[30]

S. LiuH. Yu and J.-W. Zhang, Global weak solutions of 3D compressible MHD with discontinuous initial data and vacuum, J. Differential Equations, 254 (2013), 229-255. doi: 10.1016/j.jde.2012.08.006.

[31]

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.

[32]

R. V. Polovin and V. P. Demutskii, Fundamentals Of Magnetohydrodynamics, Consultants, Bureau, New York, 1990.

[33]

A. Suen, A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density, Discrete Contin. Dyn. Syst., 33 (2013), 3791-3805. doi: 10.3934/dcds.2013.33.3791.

[34]

A. Suen and D. Hoff, Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 205 (2012), 27-58. doi: 10.1007/s00205-012-0498-3.

[35]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics 78. Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[36]

X. Xu and J. Zhang, A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum, Math. Models Methods Appl. Sci., 22 (2012), 1150010, 23 pp. doi: 10.1142/S0218202511500102.

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