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June  2016, 36(6): 3077-3106. doi: 10.3934/dcds.2016.36.3077

Global existence and optimal decay rates of solutions for compressible Hall-MHD equations

Jincheng Gao 1, and  Zheng-An Yao 1,

1. 

School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China, China

Received  April 2015 Revised  October 2015 Published  December 2015

In this paper, we are concerned with global existence and optimal decay rates of solutions for the compressible Hall-MHD equations in dimension three. First, we prove the global existence of strong solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in $H^2$-framework. Second, optimal decay rates of strong solutions in $L^2$-norm are obtained if the initial data belong to $L^1$ additionally. Finally, we apply Fourier splitting method by Schonbek [Arch. Rational Mech. Anal. 88 (1985)] to establish optimal decay rates for higher order spatial derivatives of classical solutions in $H^3$-framework, which improves the work of Fan et al.[Nonlinear Anal. Real World Appl. 22 (2015)].
Keywords: global solutions, optimal decay rates, Compressible Hall-MHD equations, Fourier splitting method..
Mathematics Subject Classification: Primary: 76W05; Secondary: 35A01, 35D35, 74H4.
Citation: Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077
References:
[1]

M. Acheritogaray, P. Degond, A. Frouvelle and J. G. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system,, Kinet. Relat. Models, 4 (2011), 901. doi: 10.3934/krm.2011.4.901. Google Scholar

[2]

S. A. Balbus and C. Terquem, Linear analysis of the Hall effect in protostellar disks,, Astrophys. J., 552 (2001), 235. doi: 10.1086/320452. Google Scholar

[3]

M. J. Benvenutti and L. C. F. Ferreira, Existence and stability of global large strong solutions for the Hall-MHD system,, preprint, (). Google Scholar

[4]

D. Chae, P. Degond and J. G. Liu, Well-posedness for Hall-magnetohydrodynamics,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555. doi: 10.1016/j.anihpc.2013.04.006. Google Scholar

[5]

D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics,, J. Differential Equations, 256 (2014), 3835. doi: 10.1016/j.jde.2014.03.003. Google Scholar

[6]

D. Chae and M. E. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations,, J. Differential Equations, 255 (2013), 3971. doi: 10.1016/j.jde.2013.07.059. Google Scholar

[7]

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. Google Scholar

[8]

R. J. Duan, H. X. Liu, S. J. Ukai and T. Yang, Optimal $L^p-L^q$ convergence rates for the compressible Navier-Stokes equations with potential force,, J. Differential Equations, 238 (2007), 220. doi: 10.1016/j.jde.2007.03.008. Google Scholar

[9]

J. S. Fan, A. Alsaedi, T. Hayat, G. Nakamura and Y. Zhou, On strong solutions to the compressible Hall-magnetohydrodynamic system,, Nonlinear Anal. Real World Appl., 22 (2015), 423. doi: 10.1016/j.nonrwa.2014.10.003. Google Scholar

[10]

J. S. Fan, F. C. Li and G. Nakamura, Regularity criteria for the incompressible Hall-magnetohydrodynamic equations,, Nonlinear Anal., 109 (2014), 173. doi: 10.1016/j.na.2014.07.003. Google Scholar

[11]

J. S. Fan and T. Ozawa, Regularity criteria for the density-dependent Hall-magnetohydrodynamics,, Appl. Math. Lett., 36 (2014), 14. doi: 10.1016/j.aml.2014.04.010. Google Scholar

[12]

J. S. Fan, S. X. Huang and G. Nakamura, Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations,, Appl. Math. Lett., 26 (2013), 963. doi: 10.1016/j.aml.2013.04.008. Google Scholar

[13]

T. G. Forbes, Magnetic reconnection in solar flares,, Geophys. Astrophys. Fluid Dyn., 62 (1991), 15. doi: 10.1080/03091929108229123. Google Scholar

[14]

J. C. Gao, Q. Tao and Z. A. Yao, Long-time behavior of solution for the compressible nematic liquid crystal flows in $\mathbbR^3$,, preprint, (). Google Scholar

[15]

J. C. Gao, Y. H. Chen and Z. A. Yao, Long-time behavior of solution to the compressible magnetohydrodynamic equations,, Nonlinear Anal., 128 (2015), 122. doi: 10.1016/j.na.2015.07.028. Google Scholar

[16]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces,, Comm. Partial Differential Equations, 37 (2012), 2165. doi: 10.1080/03605302.2012.696296. Google Scholar

[17]

H. Homann and R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems,, Phys. D, 208 (2005), 59. doi: 10.1016/j.physd.2005.06.003. Google Scholar

[18]

X. P. Hu and G. C. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows,, SIAM J. Math. Anal., 45 (2013), 2815. doi: 10.1137/120892350. Google Scholar

[19]

F. C. 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. doi: 10.1017/S0308210509001632. Google Scholar

[20]

A. Matsumura and T. Nishida, The initial value problems for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., 20 (1980), 67. Google Scholar

[21]

P. D. Mininni, D. O. Gòmez and S. M. Mahajan, Dynamo action in magnetohydrodynamics and Hall magnetohydrodynamics,, Astrophys. J., 587 (2003), 472. doi: 10.1086/368181. Google Scholar

[22]

L. Nirenberg, On elliptic partial differential euations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115. Google Scholar

[23]

M. E. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 88 (1985), 209. doi: 10.1007/BF00752111. Google Scholar

[24]

D. A. Shalybkov and V. A. Urpin, The Hall effect and the decay of magnetic fields,, Astron. Astrophys., (1997), 685. Google Scholar

[25]

Z. Tan and H. Q. Wang, Optimal decay rates of the compressible magnetohydrodynamic equations,, Nonlinear Anal. Real World Appl., 14 (2013), 188. doi: 10.1016/j.nonrwa.2012.05.012. Google Scholar

[26]

Y. J. Wang and Z. Tan, Global existence and optimal decay rate for the strong solutions in $H^2$ to the compressible Navier-Stokes equations,, Appl. Math. Lett., 24 (2011), 1778. doi: 10.1016/j.aml.2011.04.028. Google Scholar

[27]

W. J. Wang and W. K. Wang, Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces,, Discrete Contin. Dyn. Syst., 35 (2015), 513. doi: 10.3934/dcds.2015.35.513. Google Scholar

[28]

W. J. Wang, Large time behavior of solutions to the compressible Navier-Stokes equations with potential force,, J. Math. Anal. Appl., 423 (2015), 1448. doi: 10.1016/j.jmaa.2014.10.050. Google Scholar

[29]

M. Wardle, Star formation and the Hall effect,, Astrophys. Space Sci., 292 (2004), 317. Google Scholar

[30]

S. K. Weng, On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system,, preprint, (). Google Scholar

show all references

References:
[1]

M. Acheritogaray, P. Degond, A. Frouvelle and J. G. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system,, Kinet. Relat. Models, 4 (2011), 901. doi: 10.3934/krm.2011.4.901. Google Scholar

[2]

S. A. Balbus and C. Terquem, Linear analysis of the Hall effect in protostellar disks,, Astrophys. J., 552 (2001), 235. doi: 10.1086/320452. Google Scholar

[3]

M. J. Benvenutti and L. C. F. Ferreira, Existence and stability of global large strong solutions for the Hall-MHD system,, preprint, (). Google Scholar

[4]

D. Chae, P. Degond and J. G. Liu, Well-posedness for Hall-magnetohydrodynamics,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555. doi: 10.1016/j.anihpc.2013.04.006. Google Scholar

[5]

D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics,, J. Differential Equations, 256 (2014), 3835. doi: 10.1016/j.jde.2014.03.003. Google Scholar

[6]

D. Chae and M. E. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations,, J. Differential Equations, 255 (2013), 3971. doi: 10.1016/j.jde.2013.07.059. Google Scholar

[7]

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. Google Scholar

[8]

R. J. Duan, H. X. Liu, S. J. Ukai and T. Yang, Optimal $L^p-L^q$ convergence rates for the compressible Navier-Stokes equations with potential force,, J. Differential Equations, 238 (2007), 220. doi: 10.1016/j.jde.2007.03.008. Google Scholar

[9]

J. S. Fan, A. Alsaedi, T. Hayat, G. Nakamura and Y. Zhou, On strong solutions to the compressible Hall-magnetohydrodynamic system,, Nonlinear Anal. Real World Appl., 22 (2015), 423. doi: 10.1016/j.nonrwa.2014.10.003. Google Scholar

[10]

J. S. Fan, F. C. Li and G. Nakamura, Regularity criteria for the incompressible Hall-magnetohydrodynamic equations,, Nonlinear Anal., 109 (2014), 173. doi: 10.1016/j.na.2014.07.003. Google Scholar

[11]

J. S. Fan and T. Ozawa, Regularity criteria for the density-dependent Hall-magnetohydrodynamics,, Appl. Math. Lett., 36 (2014), 14. doi: 10.1016/j.aml.2014.04.010. Google Scholar

[12]

J. S. Fan, S. X. Huang and G. Nakamura, Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations,, Appl. Math. Lett., 26 (2013), 963. doi: 10.1016/j.aml.2013.04.008. Google Scholar

[13]

T. G. Forbes, Magnetic reconnection in solar flares,, Geophys. Astrophys. Fluid Dyn., 62 (1991), 15. doi: 10.1080/03091929108229123. Google Scholar

[14]

J. C. Gao, Q. Tao and Z. A. Yao, Long-time behavior of solution for the compressible nematic liquid crystal flows in $\mathbbR^3$,, preprint, (). Google Scholar

[15]

J. C. Gao, Y. H. Chen and Z. A. Yao, Long-time behavior of solution to the compressible magnetohydrodynamic equations,, Nonlinear Anal., 128 (2015), 122. doi: 10.1016/j.na.2015.07.028. Google Scholar

[16]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces,, Comm. Partial Differential Equations, 37 (2012), 2165. doi: 10.1080/03605302.2012.696296. Google Scholar

[17]

H. Homann and R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems,, Phys. D, 208 (2005), 59. doi: 10.1016/j.physd.2005.06.003. Google Scholar

[18]

X. P. Hu and G. C. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows,, SIAM J. Math. Anal., 45 (2013), 2815. doi: 10.1137/120892350. Google Scholar

[19]

F. C. 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. doi: 10.1017/S0308210509001632. Google Scholar

[20]

A. Matsumura and T. Nishida, The initial value problems for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., 20 (1980), 67. Google Scholar

[21]

P. D. Mininni, D. O. Gòmez and S. M. Mahajan, Dynamo action in magnetohydrodynamics and Hall magnetohydrodynamics,, Astrophys. J., 587 (2003), 472. doi: 10.1086/368181. Google Scholar

[22]

L. Nirenberg, On elliptic partial differential euations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115. Google Scholar

[23]

M. E. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 88 (1985), 209. doi: 10.1007/BF00752111. Google Scholar

[24]

D. A. Shalybkov and V. A. Urpin, The Hall effect and the decay of magnetic fields,, Astron. Astrophys., (1997), 685. Google Scholar

[25]

Z. Tan and H. Q. Wang, Optimal decay rates of the compressible magnetohydrodynamic equations,, Nonlinear Anal. Real World Appl., 14 (2013), 188. doi: 10.1016/j.nonrwa.2012.05.012. Google Scholar

[26]

Y. J. Wang and Z. Tan, Global existence and optimal decay rate for the strong solutions in $H^2$ to the compressible Navier-Stokes equations,, Appl. Math. Lett., 24 (2011), 1778. doi: 10.1016/j.aml.2011.04.028. Google Scholar

[27]

W. J. Wang and W. K. Wang, Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces,, Discrete Contin. Dyn. Syst., 35 (2015), 513. doi: 10.3934/dcds.2015.35.513. Google Scholar

[28]

W. J. Wang, Large time behavior of solutions to the compressible Navier-Stokes equations with potential force,, J. Math. Anal. Appl., 423 (2015), 1448. doi: 10.1016/j.jmaa.2014.10.050. Google Scholar

[29]

M. Wardle, Star formation and the Hall effect,, Astrophys. Space Sci., 292 (2004), 317. Google Scholar

[30]

S. K. Weng, On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system,, preprint, (). Google Scholar

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