# American Institute of Mathematical Sciences

October  2015, 35(10): 5083-5105. doi: 10.3934/dcds.2015.35.5083

## Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity

 1 School of Mathematical Sciences and Fujian Provincial Key Laboratory, on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen, 361005, China 2 School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005 3 School of Mathematical Sciences, Xiamen University, Fujian 361005

Received  July 2014 Revised  February 2015 Published  April 2015

In this paper, the compressible magnetohydrodynamic equations without heat conductivity are considered in $\mathbb{R}^3$. The global solution is obtained by combining the local existence and a priori estimates under the smallness assumption on the initial perturbation in $H^l (l>3)$. But we don't need the bound of $L^1$ norm. This is different from the work [5]. Our proof is based on pure estimates to get the time decay estimates on the pressure, velocity and magnet field. In particular, we use a fast decay of velocity gradient to get the uniform bound of the non-dissipative entropy, which is sufficient to close the priori estimates. In addition, we study the optimal convergence rates of the global solution.
Citation: 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
##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Anaiysis and Nonliner Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer Verlag, 2011. doi: 10.1007/978-3-642-16830-7. [2] Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Anal., 72 (2010), 4438-4451. doi: 10.1016/j.na.2010.02.019. [3] G. Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376. doi: 10.1006/jdeq.2001.4111. [4] G. Q. Chen and D. 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. [5] R. J. Duan and H. F. Ma, Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity, Indiana Univ. Math. J., 57 (2008), 2299-2319. doi: 10.1512/iumj.2008.57.3326. [6] R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rate for the compressible Navier-Stokes equations with potential force, Math. Models Methods Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X. [7] J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660. doi: 10.1016/j.na.2007.10.005. [8] J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. RWA, 10 (2009), 392-409. doi: 10.1016/j.nonrwa.2007.10.001. [9] Z. S. Gao, Z. Tan and G. C. Wu, Global existence and convergence rates of smooth solutions for 3-D the compressible magnetohydrodynamic equations without heat conductivity, Acta Mthematica Scientia, 34 (2014), 93-106. doi: 10.1016/S0252-9602(13)60129-0. [10] 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. [11] X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamics flows, Comm. Math. Phys., 283 (2008), 255-284. doi: 10.1007/s00220-008-0497-2. [12] X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamics flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9. [13] N. Ju, Existence and uniqueness of the solution to the dissipative 2D Quasi-Geostrophic equations in the Sobolev space, Comm. Math. Phys., 251 (2004), 365-376. doi: 10.1007/s00220-004-1062-2. [14] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph.D thesis, Kyoto University, 1983. [15] S. Kawashima, Smooth global solutions for two-dimensinal equations of electromagneto-fluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222. doi: 10.1007/BF03167869. [16] 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. [17] T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbb{R}^3$, Comm. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543. [18] 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. [19] F. Li and H. J. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 109-126. [20] T. P. Liu and Y. Zeng, Compressible Navier-Stokes equations with zero heat conductivity, J. Differential Equations, 153 (1999), 225-291. doi: 10.1006/jdeq.1998.3554. [21] G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X. [22] Z. Tan and H. Q. Wang, Optimal decay rates of the compressible magnetohydrodynamic equations, Nonlinear Anal. RWA, 14 (2013), 188-201. doi: 10.1016/j.nonrwa.2012.05.012. [23] Z. Tan and J. Y. Wang, On hyperbolic-dissipative systems of composite type, preprint. [24] T. Umeda, S. Kawashiwa 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. [25] A. I. Volpert and S. I. Khudiaev, On the Cauchy problem for composite systems of non-linear equations, Mat. Sb., 87 (1972), 504-528. [26] D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441. doi: 10.1137/S0036139902409284. [27] Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279. doi: 10.1007/s002050050188.

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##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Anaiysis and Nonliner Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer Verlag, 2011. doi: 10.1007/978-3-642-16830-7. [2] Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Anal., 72 (2010), 4438-4451. doi: 10.1016/j.na.2010.02.019. [3] G. Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376. doi: 10.1006/jdeq.2001.4111. [4] G. Q. Chen and D. 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. [5] R. J. Duan and H. F. Ma, Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity, Indiana Univ. Math. J., 57 (2008), 2299-2319. doi: 10.1512/iumj.2008.57.3326. [6] R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rate for the compressible Navier-Stokes equations with potential force, Math. Models Methods Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X. [7] J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660. doi: 10.1016/j.na.2007.10.005. [8] J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. RWA, 10 (2009), 392-409. doi: 10.1016/j.nonrwa.2007.10.001. [9] Z. S. Gao, Z. Tan and G. C. Wu, Global existence and convergence rates of smooth solutions for 3-D the compressible magnetohydrodynamic equations without heat conductivity, Acta Mthematica Scientia, 34 (2014), 93-106. doi: 10.1016/S0252-9602(13)60129-0. [10] 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. [11] X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamics flows, Comm. Math. Phys., 283 (2008), 255-284. doi: 10.1007/s00220-008-0497-2. [12] X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamics flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9. [13] N. Ju, Existence and uniqueness of the solution to the dissipative 2D Quasi-Geostrophic equations in the Sobolev space, Comm. Math. Phys., 251 (2004), 365-376. doi: 10.1007/s00220-004-1062-2. [14] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph.D thesis, Kyoto University, 1983. [15] S. Kawashima, Smooth global solutions for two-dimensinal equations of electromagneto-fluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222. doi: 10.1007/BF03167869. [16] 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. [17] T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbb{R}^3$, Comm. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543. [18] 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. [19] F. Li and H. J. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 109-126. [20] T. P. Liu and Y. Zeng, Compressible Navier-Stokes equations with zero heat conductivity, J. Differential Equations, 153 (1999), 225-291. doi: 10.1006/jdeq.1998.3554. [21] G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X. [22] Z. Tan and H. Q. Wang, Optimal decay rates of the compressible magnetohydrodynamic equations, Nonlinear Anal. RWA, 14 (2013), 188-201. doi: 10.1016/j.nonrwa.2012.05.012. [23] Z. Tan and J. Y. Wang, On hyperbolic-dissipative systems of composite type, preprint. [24] T. Umeda, S. Kawashiwa 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. [25] A. I. Volpert and S. I. Khudiaev, On the Cauchy problem for composite systems of non-linear equations, Mat. Sb., 87 (1972), 504-528. [26] D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441. doi: 10.1137/S0036139902409284. [27] Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279. doi: 10.1007/s002050050188.
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