# American Institute of Mathematical Sciences

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] Q.-L. Chen, C.-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. Google Scholar [6] R. Danchin, Global existence in critical spaces for compressible navier-stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] 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. Google Scholar [13] E. Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. Google Scholar [14] E. Feireisl, A. 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. Google Scholar [15] E. Feireisl, A. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] 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. Google Scholar [21] 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. Google Scholar [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. Google Scholar [23] A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, Massachusetts, 1965.Google Scholar [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. Google Scholar [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. Google Scholar [26] H.-L. Li, X.-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. Google Scholar [27] X.-L. Li, N. 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. Google Scholar [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. Google Scholar [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. Google Scholar [30] S. Liu, H. 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. Google Scholar [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. Google Scholar [32] R. V. Polovin and V. P. Demutskii, Fundamentals Of Magnetohydrodynamics, Consultants, Bureau, New York, 1990.Google Scholar [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. Google Scholar [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. Google Scholar [35] H. Triebel, Theory of Function Spaces, Monographs in Mathematics 78. Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] Q.-L. Chen, C.-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. Google Scholar [6] R. Danchin, Global existence in critical spaces for compressible navier-stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] 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. Google Scholar [13] E. Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. Google Scholar [14] E. Feireisl, A. 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. Google Scholar [15] E. Feireisl, A. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] 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. Google Scholar [21] 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. Google Scholar [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. Google Scholar [23] A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, Massachusetts, 1965.Google Scholar [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. Google Scholar [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. Google Scholar [26] H.-L. Li, X.-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. Google Scholar [27] X.-L. Li, N. 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. Google Scholar [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. Google Scholar [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. Google Scholar [30] S. Liu, H. 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. Google Scholar [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. Google Scholar [32] R. V. Polovin and V. P. Demutskii, Fundamentals Of Magnetohydrodynamics, Consultants, Bureau, New York, 1990.Google Scholar [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. Google Scholar [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. Google Scholar [35] H. Triebel, Theory of Function Spaces, Monographs in Mathematics 78. Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1. Google Scholar [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. Google Scholar
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