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December  2017, 10(4): 1035-1053. doi: 10.3934/krm.2017041

Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum

1. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China

2. 

School of Mathematics, Shandong University, Jinan 250100, China

3. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

* S. Huang is the corresponding author

Received  September 2015 Revised  November 2016 Published  March 2017

This paper considers the initial boundary problem to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. The global existence and uniqueness of large strong solutions are established when the heat conductivity coefficient
$κ(θ)$
satisfies
$C_{1}(1+\theta^q)\leq \kappa(\theta)\leq C_2(1+\theta^q)$
for some constants
$q>0$
, and
$C_1,C_2>0$
.
Citation: Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic & Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041
References:
[1]

A. A. Amosov and A. A. Zlotnik, Global generalized solutions of the equations of the one-dimensional motion of a viscous heat-conducting gas, Sov. Mat. Dokl., 38 (1989), 1-5.   Google Scholar

[2]

D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., 87 (2007), 57-90.  doi: 10.1016/j.matpur.2006.11.001.  Google Scholar

[3]

H. Cabannes, Theoretical Magnetofluiddynamics Academic Press, New York, London, 1970. Google Scholar

[4]

G.-Q. Chen and D.-H. Wang, Globla solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.  doi: 10.1006/jdeq.2001.4111.  Google Scholar

[5]

G.-Q. Chen and D.-H. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamics equations, Z. Angew. Math. Phys., 54 (2003), 608-632.  doi: 10.1007/s00033-003-1017-z.  Google Scholar

[6]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[7]

B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.  Google Scholar

[8]

J.-S. FanS. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys., 270 (2007), 691-708.  doi: 10.1007/s00220-006-0167-1.  Google Scholar

[9]

J.-S. FanS. 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.  Google Scholar

[10]

J.-S. Fan and W.-H. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660.  doi: 10.1016/j.na.2007.10.005.  Google Scholar

[11]

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.  Google Scholar

[12]

E. Feireisl, Dynamics Of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26 Oxford University Press, Oxford, 2004.  Google Scholar

[13]

D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804.  doi: 10.1007/s00033-005-4057-8.  Google Scholar

[14]

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

[15]

X.-P. Hu and D.-H. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2.  Google Scholar

[16]

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.  Google Scholar

[17]

Y.-X. Hu and Q.-C. Ju, Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity, Z. Angew. Math. Phys., 66 (2015), 865-889.  doi: 10.1007/s00033-014-0446-1.  Google Scholar

[18]

X. -D. Huang and J. Li, Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier-Stokes System with Vacuum and Large Oscillations arXiv: 1107.4655v2, [math. ph]. Google Scholar

[19]

A. Jeffrey and T. Taniuti, Non-Linear Wave Propagation. With Applications To Physics And Magnetohydrodynamics, Academic Press, New York, 1964.  Google Scholar

[20]

S. Kawashima, Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.  doi: 10.1007/BF03167869.  Google Scholar

[21]

S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 384-387.  doi: 10.3792/pjaa.58.384.  Google Scholar

[22]

B. Kawohl, Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103.  doi: 10.1016/0022-0396(85)90023-3.  Google Scholar

[23]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initialboundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech. , 41 (1977), 273–282, translated from Prikl. Mat. Meh. , 41 (1977), 282–291 (Russian). doi: 10.1016/0021-8928(77)90011-9.  Google Scholar

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[28]

T.-P Liu and Y. Zeng, Large-time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997), viii+120 pp.  doi: 10.1090/memo/0599.  Google Scholar

[29]

R.-H. Pan and W.-Z. Zhang, Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401-425.  doi: 10.4310/CMS.2015.v13.n2.a7.  Google Scholar

[30]

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

[31]

G. Ströhmer, About compressible viscous fluid flow in a bounded region, Pacific J. Math., 143 (1990), 359-375.  doi: 10.2140/pjm.1990.143.359.  Google Scholar

[32]

A. Suen and D. Hoff, About compressible viscous fluid flow in a bounded region, Pacific J. Math., 143 (1990), 359-375.  doi: 10.1007/s00205-012-0498-3.  Google Scholar

[33]

E. Tsyganov and D. Hoff, Systems of partial differential equations of mixed hyperbolic-parabolic type, J. Differential Equations, 204 (2004), 163-201.  doi: 10.1016/j.jde.2004.03.010.  Google Scholar

[34]

T. UmedaS. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.  doi: 10.1007/BF03167068.  Google Scholar

[35]

A. I. Vol'pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, (Russian), Mat. Sb. (N.S.), 87(129) (1972), 504-528.   Google Scholar

[36]

D.-H. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441.  doi: 10.1137/S0036139902409284.  Google Scholar

[37]

H. Wen and C. Zhu, Global classical large solutions to Navier-Stokes equations for viscous compressible and heat conducting fluids with vacuum, SIAM J. Math. Anal., 45 (2013), 431-468.  doi: 10.1137/120877829.  Google Scholar

[38]

H.-Y. Wen and C.-J. Zhu, Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data, J. Math. Pures Appl.(9), 102 (2014), 498-545.  doi: 10.1016/j.matpur.2013.12.003.  Google Scholar

[39]

J.-W. ZhangS. Jiang and F. Xie, Global weak solutions of an initial boundary value problem for screw pinches in plasma physics, Math. Models Methods Appl. Sci., 19 (2009), 833-875.  doi: 10.1142/S0218202509003644.  Google Scholar

[40]

A. A. Zlotnik and A. A. Amosov, On stability of generalized solutions to the equations of one-dimensional motion of a viscous heat-conducting gas, Sib. Math. J., 38 (1997), 663-684.  doi: 10.1007/BF02674573.  Google Scholar

[41]

A. A. Zlotnik and A. A. Amosov, Stability of generalized solutions to equations of one-dimensional motion of viscous heat conducting gases, Math. Notes, 63 (1998), 736-746.  doi: 10.1007/BF02312766.  Google Scholar

show all references

References:
[1]

A. A. Amosov and A. A. Zlotnik, Global generalized solutions of the equations of the one-dimensional motion of a viscous heat-conducting gas, Sov. Mat. Dokl., 38 (1989), 1-5.   Google Scholar

[2]

D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., 87 (2007), 57-90.  doi: 10.1016/j.matpur.2006.11.001.  Google Scholar

[3]

H. Cabannes, Theoretical Magnetofluiddynamics Academic Press, New York, London, 1970. Google Scholar

[4]

G.-Q. Chen and D.-H. Wang, Globla solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.  doi: 10.1006/jdeq.2001.4111.  Google Scholar

[5]

G.-Q. Chen and D.-H. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamics equations, Z. Angew. Math. Phys., 54 (2003), 608-632.  doi: 10.1007/s00033-003-1017-z.  Google Scholar

[6]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[7]

B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.  Google Scholar

[8]

J.-S. FanS. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys., 270 (2007), 691-708.  doi: 10.1007/s00220-006-0167-1.  Google Scholar

[9]

J.-S. FanS. 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.  Google Scholar

[10]

J.-S. Fan and W.-H. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660.  doi: 10.1016/j.na.2007.10.005.  Google Scholar

[11]

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.  Google Scholar

[12]

E. Feireisl, Dynamics Of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26 Oxford University Press, Oxford, 2004.  Google Scholar

[13]

D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804.  doi: 10.1007/s00033-005-4057-8.  Google Scholar

[14]

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

[15]

X.-P. Hu and D.-H. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2.  Google Scholar

[16]

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.  Google Scholar

[17]

Y.-X. Hu and Q.-C. Ju, Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity, Z. Angew. Math. Phys., 66 (2015), 865-889.  doi: 10.1007/s00033-014-0446-1.  Google Scholar

[18]

X. -D. Huang and J. Li, Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier-Stokes System with Vacuum and Large Oscillations arXiv: 1107.4655v2, [math. ph]. Google Scholar

[19]

A. Jeffrey and T. Taniuti, Non-Linear Wave Propagation. With Applications To Physics And Magnetohydrodynamics, Academic Press, New York, 1964.  Google Scholar

[20]

S. Kawashima, Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.  doi: 10.1007/BF03167869.  Google Scholar

[21]

S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 384-387.  doi: 10.3792/pjaa.58.384.  Google Scholar

[22]

B. Kawohl, Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103.  doi: 10.1016/0022-0396(85)90023-3.  Google Scholar

[23]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initialboundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech. , 41 (1977), 273–282, translated from Prikl. Mat. Meh. , 41 (1977), 282–291 (Russian). doi: 10.1016/0021-8928(77)90011-9.  Google Scholar

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[28]

T.-P Liu and Y. Zeng, Large-time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997), viii+120 pp.  doi: 10.1090/memo/0599.  Google Scholar

[29]

R.-H. Pan and W.-Z. Zhang, Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401-425.  doi: 10.4310/CMS.2015.v13.n2.a7.  Google Scholar

[30]

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

[31]

G. Ströhmer, About compressible viscous fluid flow in a bounded region, Pacific J. Math., 143 (1990), 359-375.  doi: 10.2140/pjm.1990.143.359.  Google Scholar

[32]

A. Suen and D. Hoff, About compressible viscous fluid flow in a bounded region, Pacific J. Math., 143 (1990), 359-375.  doi: 10.1007/s00205-012-0498-3.  Google Scholar

[33]

E. Tsyganov and D. Hoff, Systems of partial differential equations of mixed hyperbolic-parabolic type, J. Differential Equations, 204 (2004), 163-201.  doi: 10.1016/j.jde.2004.03.010.  Google Scholar

[34]

T. UmedaS. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.  doi: 10.1007/BF03167068.  Google Scholar

[35]

A. I. Vol'pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, (Russian), Mat. Sb. (N.S.), 87(129) (1972), 504-528.   Google Scholar

[36]

D.-H. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441.  doi: 10.1137/S0036139902409284.  Google Scholar

[37]

H. Wen and C. Zhu, Global classical large solutions to Navier-Stokes equations for viscous compressible and heat conducting fluids with vacuum, SIAM J. Math. Anal., 45 (2013), 431-468.  doi: 10.1137/120877829.  Google Scholar

[38]

H.-Y. Wen and C.-J. Zhu, Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data, J. Math. Pures Appl.(9), 102 (2014), 498-545.  doi: 10.1016/j.matpur.2013.12.003.  Google Scholar

[39]

J.-W. ZhangS. Jiang and F. Xie, Global weak solutions of an initial boundary value problem for screw pinches in plasma physics, Math. Models Methods Appl. Sci., 19 (2009), 833-875.  doi: 10.1142/S0218202509003644.  Google Scholar

[40]

A. A. Zlotnik and A. A. Amosov, On stability of generalized solutions to the equations of one-dimensional motion of a viscous heat-conducting gas, Sib. Math. J., 38 (1997), 663-684.  doi: 10.1007/BF02674573.  Google Scholar

[41]

A. A. Zlotnik and A. A. Amosov, Stability of generalized solutions to equations of one-dimensional motion of viscous heat conducting gases, Math. Notes, 63 (1998), 736-746.  doi: 10.1007/BF02312766.  Google Scholar

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