# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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, (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. 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. 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. 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. 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. doi: 10.1142/S021820250700208X. [7] J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations,, Nonlinear Anal., 69 (2008), 3637. 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. 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. 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. 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. 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. 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. 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, (1983). [15] S. Kawashima, Smooth global solutions for two-dimensinal equations of electromagneto-fluid dynamics,, Japan J. Appl. Math., 1 (1984), 207. 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. 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 $\mathbbR^3$,, Comm. Math. Phys., 200 (1999), 621. 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. 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. [20] T. P. Liu and Y. Zeng, Compressible Navier-Stokes equations with zero heat conductivity,, J. Differential Equations, 153 (1999), 225. 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. 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. 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. 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. [26] D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics,, SIAM J. Appl. Math., 63 (2003), 1424. 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. 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, (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. 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. 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. 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. 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. doi: 10.1142/S021820250700208X. [7] J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations,, Nonlinear Anal., 69 (2008), 3637. 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. 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. 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. 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. 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. 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. 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, (1983). [15] S. Kawashima, Smooth global solutions for two-dimensinal equations of electromagneto-fluid dynamics,, Japan J. Appl. Math., 1 (1984), 207. 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. 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 $\mathbbR^3$,, Comm. Math. Phys., 200 (1999), 621. 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. 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. [20] T. P. Liu and Y. Zeng, Compressible Navier-Stokes equations with zero heat conductivity,, J. Differential Equations, 153 (1999), 225. 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. 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. 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. 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. [26] D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics,, SIAM J. Appl. Math., 63 (2003), 1424. 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. doi: 10.1007/s002050050188.
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