July  2016, 21(5): 1617-1633. doi: 10.3934/dcdsb.2016014

Local strong solutions to the compressible viscous magnetohydrodynamic equations

1. 

Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098

2. 

Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, School of Mathematical Science, Nanjing Normal University, Nanjing 210023

Received  January 2014 Revised  March 2014 Published  April 2016

In this paper, we consider the compressible magnetohydrodynamic equations with nonnegative thermal conductivity and electric conductivity. The coefficients of the viscosity, heat conductivity and magnetic diffusivity depend on density and temperature. Inspired by the framework of [11], [13] and [15], we use the maximal regularity and contraction mapping argument to prove the existence and uniqueness of local strong solutions with positive initial density in the bounded domain for any dimension.
Citation: Tong Tang, Hongjun Gao. Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1617-1633. doi: 10.3934/dcdsb.2016014
References:
[1]

C. S. Cao and J. H. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.

[2]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114 pp. doi: 10.1090/memo/0788.

[3]

R. Denk, M. Hieber and J. Prüss, Optimal $L_p-L_q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.

[4]

J. S. Fan and W. H. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. doi: 10.1016/j.nonrwa.2007.10.001.

[5]

J. S. Fan, H. J. Gao and G. Nakamura, Regularity criteria for the generalized magnetohydrodynamic equations and the quasi-geostrophic equations, Taiwanese J. Math., 15 (2011), 1059-1073.

[6]

J. S. Fan, S. Jiang and G. Nakamura, Stability of weak solutions to equations of magnetohydrodynamics with Lebesgue initial data, J. Differential Equations, 251 (2011), 2025-2036. doi: 10.1016/j.jde.2011.06.019.

[7]

X. P. Hu and D. H. Wang, Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations, J. Differential Equations, 245 (2008), 2176-2198. doi: 10.1016/j.jde.2008.07.019.

[8]

X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382.

[9]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543.

[10]

S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal., 12 (2005), 239-251. doi: 10.4310/MAA.2005.v12.n3.a2.

[11]

T. Kato, Quasi-linear Equations Of Evolution with Applications to Partial Differential Equations, Springer, 1974.

[12]

M. Kotschote, Strong solutions to the Navier-Stokes equations for a compressible fluid of Allen-Cahn type, Arch. Rational Mech. Anal., 206 (2012), 489-514. doi: 10.1007/s00205-012-0538-z.

[13]

M. Kotschote, Strong solutions to the compressible non-isothermal Navier-Stokes equations, Adv.Math.Sci.Appl., 22 (2012), 319-347.

[14]

M. Kotschote and R. Zacher, Strong solutions in the dynamical theory of compressible two phase fluids, arXiv:1306.2565v1.

[15]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, 1973.

[16]

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.

[17]

P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models, Oxford University Press, New York, 1998.

[18]

A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[19]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.

[20]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738.

[21]

J. Prüss, Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in $L_p$-spaces, Math. Bohem., 127 (2002), 311-327.

[22]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203 (1990), 429-452. doi: 10.1007/BF02570748.

[23]

X. K. Pu and B. L. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538. doi: 10.1007/s00033-012-0245-5.

[24]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.

[25]

J. H. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306. doi: 10.1080/03605300701382530.

[26]

J. H. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.

show all references

References:
[1]

C. S. Cao and J. H. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.

[2]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114 pp. doi: 10.1090/memo/0788.

[3]

R. Denk, M. Hieber and J. Prüss, Optimal $L_p-L_q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.

[4]

J. S. Fan and W. H. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. doi: 10.1016/j.nonrwa.2007.10.001.

[5]

J. S. Fan, H. J. Gao and G. Nakamura, Regularity criteria for the generalized magnetohydrodynamic equations and the quasi-geostrophic equations, Taiwanese J. Math., 15 (2011), 1059-1073.

[6]

J. S. Fan, S. Jiang and G. Nakamura, Stability of weak solutions to equations of magnetohydrodynamics with Lebesgue initial data, J. Differential Equations, 251 (2011), 2025-2036. doi: 10.1016/j.jde.2011.06.019.

[7]

X. P. Hu and D. H. Wang, Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations, J. Differential Equations, 245 (2008), 2176-2198. doi: 10.1016/j.jde.2008.07.019.

[8]

X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382.

[9]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543.

[10]

S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal., 12 (2005), 239-251. doi: 10.4310/MAA.2005.v12.n3.a2.

[11]

T. Kato, Quasi-linear Equations Of Evolution with Applications to Partial Differential Equations, Springer, 1974.

[12]

M. Kotschote, Strong solutions to the Navier-Stokes equations for a compressible fluid of Allen-Cahn type, Arch. Rational Mech. Anal., 206 (2012), 489-514. doi: 10.1007/s00205-012-0538-z.

[13]

M. Kotschote, Strong solutions to the compressible non-isothermal Navier-Stokes equations, Adv.Math.Sci.Appl., 22 (2012), 319-347.

[14]

M. Kotschote and R. Zacher, Strong solutions in the dynamical theory of compressible two phase fluids, arXiv:1306.2565v1.

[15]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, 1973.

[16]

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.

[17]

P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models, Oxford University Press, New York, 1998.

[18]

A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[19]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.

[20]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738.

[21]

J. Prüss, Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in $L_p$-spaces, Math. Bohem., 127 (2002), 311-327.

[22]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203 (1990), 429-452. doi: 10.1007/BF02570748.

[23]

X. K. Pu and B. L. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538. doi: 10.1007/s00033-012-0245-5.

[24]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.

[25]

J. H. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306. doi: 10.1080/03605300701382530.

[26]

J. H. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.

[1]

Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic and Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041

[2]

Qing Chen, Zhong Tan. Global existence in critical spaces for the compressible magnetohydrodynamic equations. Kinetic and Related Models, 2012, 5 (4) : 743-767. doi: 10.3934/krm.2012.5.743

[3]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091

[4]

Yu-Zhu Wang, Yin-Xia Wang. Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity. Communications on Pure and Applied Analysis, 2013, 12 (2) : 851-866. doi: 10.3934/cpaa.2013.12.851

[5]

Zhong Tan, Qiuju Xu, Huaqiao Wang. Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5083-5105. doi: 10.3934/dcds.2015.35.5083

[6]

Hong Cai, Zhong Tan. Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1847-1868. doi: 10.3934/dcds.2016.36.1847

[7]

Eduard Feireisl, Antonin Novotny, Yongzhong Sun. Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 121-143. doi: 10.3934/dcds.2014.34.121

[8]

Tong Tang, Yongfu Wang. Strong solutions to compressible barotropic viscoelastic flow with vacuum. Kinetic and Related Models, 2015, 8 (4) : 765-775. doi: 10.3934/krm.2015.8.765

[9]

Jishan Fan, Fucai Li, Gen Nakamura. Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain. Conference Publications, 2015, 2015 (special) : 387-394. doi: 10.3934/proc.2015.0387

[10]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic and Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481

[11]

Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4515-4533. doi: 10.3934/dcdsb.2020110

[12]

G. Deugoué, J. K. Djoko, A. C. Fouape, A. Ndongmo Ngana. Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamic equations. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1509-1535. doi: 10.3934/cpaa.2020076

[13]

Jianjun Chen, Geng Lai. Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations. Communications on Pure and Applied Analysis, 2019, 18 (2) : 943-958. doi: 10.3934/cpaa.2019046

[14]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553

[15]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337

[16]

Yanmin Mu. Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces. Kinetic and Related Models, 2014, 7 (4) : 739-753. doi: 10.3934/krm.2014.7.739

[17]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5383-5405. doi: 10.3934/dcdsb.2020348

[18]

Geonho Lee, Sangdong Kim, Young-Sam Kwon. Large time behavior for the full compressible magnetohydrodynamic flows. Communications on Pure and Applied Analysis, 2012, 11 (3) : 959-971. doi: 10.3934/cpaa.2012.11.959

[19]

Weiping Yan. Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1359-1385. doi: 10.3934/dcds.2015.35.1359

[20]

Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]