On global solutions and asymptotic behavior of planar magnetohydrodynamics with large data
Author:
Yuxi Hu
Journal:
Quart. Appl. Math. 73 (2015), 759-772
MSC (2010):
Primary 35B40, 35Q35, 76N10
DOI:
https://doi.org/10.1090/qam/1413
Published electronically:
September 11, 2015
MathSciNet review:
3432282
Full-text PDF Free Access
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Abstract: In this paper we consider an initial boundary value problem for planar magnetohydrodynamic compressible flows. By assuming that the adiabatic constant $\gamma$ is sufficiently close to $1$, we prove the existence and uniqueness of global strong solutions with large initial data when all the viscosity, heat conductivity, and diffusivity coefficients are constant. Moreover, the asymptotic behavior of solutions is also investigated.
References
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- H. Cabannes, Theoretical magnetofluiddynamics, Academic Press, New York, 1970.
- Gui-Qiang Chen and Dehua Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations 182 (2002), no. 2, 344–376. MR 1900327, DOI https://doi.org/10.1006/jdeq.2001.4111
- Gui-Qiang Chen and Dehua Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys. 54 (2003), no. 4, 608–632. MR 1994028, DOI https://doi.org/10.1007/s00033-003-1017-z
- P. C. Clemmow and J. P. Dougherty, Electrodynamics of particles and plasmas, Addison-Wesley, New York, 1990.
- J. S. Fan, S. X. Huang, and F. C. Li, Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vaccum, preprint (2013).
- Jishan Fan, Song Jiang, and Gen Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys. 270 (2007), no. 3, 691–708. MR 2276461, DOI https://doi.org/10.1007/s00220-006-0167-1
- H. Freistühler and P. Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal. 26 (1995), no. 1, 112–128. MR 1311884, DOI https://doi.org/10.1137/S0036141093247366
- David Hoff and Eugene Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys. 56 (2005), no. 5, 791–804. MR 2184906, DOI https://doi.org/10.1007/s00033-005-4057-8
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- Yuxi Hu and Qiangchang Ju, Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity, Z. Angew. Math. Phys. 66 (2015), no. 3, 865–889. MR 3347415, DOI https://doi.org/10.1007/s00033-014-0446-1
- A. Jeffrey and T. Taniuti, Non-linear wave propagation. With applications to physics and magnetohydrodynamics, Academic Press, New York-London, 1964. MR 0167137
- Song Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z. 216 (1994), no. 2, 317–336. MR 1278427, DOI https://doi.org/10.1007/BF02572324
- Shuichi Kawashima and Takaaki Nishida, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. Math. Kyoto Univ. 21 (1981), no. 4, 825–837. MR 637519, DOI https://doi.org/10.1215/kjm/1250521915
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- Shuichi Kawashima and Yasushi Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math. 10 (1986), no. 1, 131–149. MR 846424, DOI https://doi.org/10.21099/tkbjm/1496160397
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- A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, Massachusetts, 1965.
- L. D. Laudau and E. M. Lifshitz, Electrodynamics of continuous media, 2nd ed., Pergamon, New York, 1984.
- Hongxia Liu, Tong Yang, Huijiang Zhao, and Qingyang Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data, SIAM J. Math. Anal. 46 (2014), no. 3, 2185–2228. MR 3225502, DOI https://doi.org/10.1137/130920617
- Tai-Ping Liu and Yanni Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc. 125 (1997), no. 599, viii+120. MR 1357824, DOI https://doi.org/10.1090/memo/0599
- John Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général, Bull. Soc. Math. France 90 (1962), 487–497 (French). MR 149094
- Mari Okada and Shuichi Kawashima, On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ. 23 (1983), no. 1, 55–71. MR 692729, DOI https://doi.org/10.1215/kjm/1250521610
- R. V. Polovin and V. P. Demutskii, Fundamentals of magnetohydrodynamics, Consultants Bureau, New York, 1990.
- Zhong Tan, Tong Yang, Huijiang Zhao, and Qingyang Zou, Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data, SIAM J. Math. Anal. 45 (2013), no. 2, 547–571. MR 3032988, DOI https://doi.org/10.1137/120876174
- A. I. Vol′pert and S. I. Hudjaev, The Cauchy problem for composite systems of nonlinear differential equations, Mat. Sb. (N.S.) 87(129) (1972), 504–528 (Russian). MR 0390528
- Dehua Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math. 63 (2003), no. 4, 1424–1441. MR 1989910, DOI https://doi.org/10.1137/S0036139902409284
- L. C. Woods, Principles of magnetoplasma dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1987. MR 957923
- C. C. Wu, Formation, structure, and stability of MHD intermediate shocks, J. Geophys. Res. 95 (1990), 8149-8175.
References
- R. Balescu, Transport processes in plasmas I: Classical transport theory, North-Holland, 1988.
- H. Cabannes, Theoretical magnetofluiddynamics, Academic Press, New York, 1970.
- Gui-Qiang Chen and Dehua Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations 182 (2002), no. 2, 344–376. MR 1900327 (2003d:35212), DOI https://doi.org/10.1006/jdeq.2001.4111
- Gui-Qiang Chen and Dehua Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys. 54 (2003), no. 4, 608–632. MR 1994028 (2005b:35227), DOI https://doi.org/10.1007/s00033-003-1017-z
- P. C. Clemmow and J. P. Dougherty, Electrodynamics of particles and plasmas, Addison-Wesley, New York, 1990.
- J. S. Fan, S. X. Huang, and F. C. Li, Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vaccum, preprint (2013).
- Jishan Fan, Song Jiang, and Gen Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys. 270 (2007), no. 3, 691–708. MR 2276461 (2008g:76135), DOI https://doi.org/10.1007/s00220-006-0167-1
- H. Freistühler and P. Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal. 26 (1995), no. 1, 112–128. MR 1311884 (95j:35183), DOI https://doi.org/10.1137/S0036141093247366
- David Hoff and Eugene Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys. 56 (2005), no. 5, 791–804. MR 2184906 (2006k:35230), DOI https://doi.org/10.1007/s00033-005-4057-8
- Xianpeng Hu and Dehua Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 203–238. MR 2646819 (2011d:35079), DOI https://doi.org/10.1007/s00205-010-0295-9
- Yuxi Hu and Qiangchang Ju, Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity, Z. Angew. Math. Phys. 66 (2015), no. 3, 865–889. MR 3347415, DOI https://doi.org/10.1007/s00033-014-0446-1
- A. Jeffrey and T. Taniuti, Non-linear wave propagation. With applications to physics and magnetohydrodynamics, Academic Press, New York-London, 1964. MR 0167137 (29 \#4410)
- Song Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z. 216 (1994), no. 2, 317–336. MR 1278427 (95e:35120), DOI https://doi.org/10.1007/BF02572324
- Shuichi Kawashima and Takaaki Nishida, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. Math. Kyoto Univ. 21 (1981), no. 4, 825–837. MR 637519 (84d:76042)
- Shuichi Kawashima and Mari Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 9, 384–387. MR 694940 (85c:35078)
- Shuichi Kawashima and Yasushi Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math. 10 (1986), no. 1, 131–149. MR 846424 (87m:76070a)
- A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, Prikl. Mat. Meh. 41 (1977), no. 2, 282–291 (Russian); English transl., J. Appl. Math. Mech. 41 (1977), no. 2, 273–282. MR 0468593 (57 \#8425)
- A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, Massachusetts, 1965.
- L. D. Laudau and E. M. Lifshitz, Electrodynamics of continuous media, 2nd ed., Pergamon, New York, 1984.
- Hongxia Liu, Tong Yang, Huijiang Zhao, and Qingyang Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data, SIAM J. Math. Anal. 46 (2014), no. 3, 2185–2228. MR 3225502, DOI https://doi.org/10.1137/130920617
- Tai-Ping Liu and Yanni Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc. 125 (1997), no. 599, viii+120. MR 1357824 (97g:35107), DOI https://doi.org/10.1090/memo/0599
- John Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général, Bull. Soc. Math. France 90 (1962), 487–497 (French). MR 0149094 (26 \#6590)
- Mari Okada and Shuichi Kawashima, On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ. 23 (1983), no. 1, 55–71. MR 692729 (85c:76050)
- R. V. Polovin and V. P. Demutskii, Fundamentals of magnetohydrodynamics, Consultants Bureau, New York, 1990.
- Zhong Tan, Tong Yang, Huijiang Zhao, and Qingyang Zou, Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data, SIAM J. Math. Anal. 45 (2013), no. 2, 547–571. MR 3032988, DOI https://doi.org/10.1137/120876174
- A. I. Vol′pert and S. I. Hudjaev, The Cauchy problem for composite systems of nonlinear differential equations, Mat. Sb. (N.S.) 87(129) (1972), 504–528 (Russian). MR 0390528 (52 \#11353)
- Dehua Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math. 63 (2003), no. 4, 1424–1441 (electronic). MR 1989910 (2004f:76121), DOI https://doi.org/10.1137/S0036139902409284
- L. C. Woods, Principles of magnetoplasma dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1987. MR 957923 (89i:76099)
- C. C. Wu, Formation, structure, and stability of MHD intermediate shocks, J. Geophys. Res. 95 (1990), 8149-8175.
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Additional Information
Yuxi Hu
Affiliation:
Department of Mathematics, China University of Mining and Technology, Beijing 100083, People’s Republic of China
Email:
yxhu86@163.com
Received by editor(s):
April 14, 2014
Received by editor(s) in revised form:
August 17, 2014
Published electronically:
September 11, 2015
Article copyright:
© Copyright 2015
Brown University
