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.

show all references

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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