February  2021, 26(2): 987-1010. doi: 10.3934/dcdsb.2020150

Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

2. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

* Corresponding author: Xiuli Xu

Received  July 2019 Revised  December 2019 Published  February 2021 Early access  May 2020

Fund Project: The second author is supported by NSF grant 11871172 and the Natural Science Foundation of Guangdong Province of China under 2019A1515012000

In this paper, we consider the quantum magnetohydrodynamic model for quantum plasmas with potential force. We prove the optimal decay rates for the solution to the stationary state in the whole space in the $ L^{q}-L^{2} $ norm with $ 1\leq q\leq2 $. The proof is based on the optimal decay of the linearized equations, multi-frequency decompositions and nonlinear energy estimates.

Citation: Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 987-1010. doi: 10.3934/dcdsb.2020150
References:
[1]

K. Beauchard and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Rational Mech. Anal., 199 (2011), 177-227.  doi: 10.1007/s00205-010-0321-y.

[2]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamical equations, Nonl. Anal., 72 (2010), 4438-4451.  doi: 10.1016/j.na.2010.02.019.

[3]

R. DuanH. LiuS. Ukai and T. Yang, Optimal $L^p$-$L^q$ convergence rate for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233.  doi: 10.1016/j.jde.2007.03.008.

[4]

R. DuanS. UkaiT. Yang and H. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Math. Models Methods Appl. Sci., 17 (2007), 737-758.  doi: 10.1142/S021820250700208X.

[5]

J. Gao, Q. Tao and Z. Yao, Optimal decay rates of classical solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 67 (2016), Art. 23, 22 pp. doi: 10.1007/s00033-016-0616-4.

[6]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Commun. Part. Diff. Eq., 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.

[7]

F. Haas, A magnetohydrodynamic model for quantum plasmas, Phys. Plasmas, 12 (2005), 062117. doi: 10.1063/1.1939947.

[8]

F. Haas, Quantum Plasmas: An Hydrodynamic Approach, Springer, New York, 2011. doi: 10.1007/978-1-4419-8201-8.

[9]

X. Hu and D. Wang, Global solutions to the three-dimisional full compressible magnetohydroynamics flows, Comm. Math. Phys., 283 (2008), 253-284.  doi: 10.1007/s00220-008-0497-2.

[10]

X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.

[11]

N. Jiang and L. Xiong, Diffusive limit of the Boltzmann equation with fluid initial layer in the periodic domain, SIAM J. Math. Anal., 47 (2015), 1747-1777.  doi: 10.1137/130922239.

[12]

T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in the three-dimensinal exterior domain, J. Differ. Equ., 184 (2002), 587-619.  doi: 10.1006/jdeq.2002.4158.

[13]

Y. Kagei and T. Kobayashi, On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in $\Bbb R^3$, Arch. Rational Mech. Anal., 165 (2002), 89-159.  doi: 10.1007/s00205-002-0221-x.

[14]

Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Rational Mech. Anal., 177 (2005), 231-330.  doi: 10.1007/s00205-005-0365-6.

[15]

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 $\Bbb R^3$, Comm. Math. Phys., 200 (1999), 621-659.  doi: 10.1007/s002200050543.

[16]

Y. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232.  doi: 10.1016/j.jmaa.2011.11.006.

[17]

H. Liu and X. Pu, Long Wavelength limit for the quantum Euler-Poisson equation, SIAM J. Math. Anal., 48 (2016), 2345-2381.  doi: 10.1137/15M1046587.

[18]

F. Li and H. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 109-126.  doi: 10.1017/S0308210509001632.

[19]

Q. Liu and C. Zhu, Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations, Indiana Univ. Math. J., 63 (2014), 1085-1108.  doi: 10.1512/iumj.2014.63.5283.

[20]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A., 55 (1979), 337-342.  doi: 10.3792/pjaa.55.337.

[21]

A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464.  doi: 10.1007/BF01214738.

[22]

G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 339-418.  doi: 10.1016/0362-546X(85)90001-X.

[23]

X. Pu and B. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538.  doi: 10.1007/s00033-012-0245-5.

[24]

X. Pu and B. Guo, Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction, Kinet. Related Models, 9 (2016), 165-191.  doi: 10.3934/krm.2016.9.165.

[25]

X. Pu and X. Xu, Decay rates of the magnetohydrodynamic model for quantum plasmas, Z. Angew. Math. Phys., 68 (2017), Paper No. 18, 17 pp. doi: 10.1007/s00033-016-0762-8.

[26]

Z. Tan and H. Wang, Optimal decay rates of the compressible magnetohydrodynamic equations, Nonlinear Analysis: Real World Applications, 14 (2013), 188-201.  doi: 10.1016/j.nonrwa.2012.05.012.

[27]

Z. TanX. Zhang and H. Wang, Asymptotic behavior of Navier-Stokes-Korteweg with friction in $\mathbb{R}^{3}$, Discrete and Continuous Dynamical Systems, 34 (2014), 2243-2259.  doi: 10.3934/dcds.2014.34.2243.

[28]

T. UmedaS. Kawashiwa and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan. J. Appl. Math., 1 (1984), 435-457.  doi: 10.1007/BF03167068.

[29]

S. UkaiT. Yang and H. Zhao, Convergence rate for the compressible Navier-Stokes equations with external force, J. Hyperbolic Differ. Equ., 3 (2006), 561-574.  doi: 10.1142/S0219891606000902.

[30]

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.  doi: 10.1103/PhysRev.40.749.

[31]

Y. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297.  doi: 10.1016/j.jde.2012.03.006.

[32]

J. Wang, Optimal convergence rates for the strong solutions to the compressible Navier-Stokes equations with potential force, Nonlinear Anal. Real World Appl., 34 (2017), 363-378.  doi: 10.1016/j.nonrwa.2016.09.005.

[33]

Y. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271.  doi: 10.1016/j.jmaa.2011.01.006.

[34]

L. WangQ. XiaoL. Xiong and H. Zhao, The Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1049-1097.  doi: 10.1016/S0252-9602(16)30057-1.

[35]

L. Xiong, Incompressible limit of isentropic Navier-Stokes equations with Navier-slip boundary, Kinet. Relat. Models, 11 (2018), 469-490.  doi: 10.3934/krm.2018021.

[36]

X. Xi, X. Pu and B. Guo, Long-time behavior of solutions for the compressible quantum magnetohydrodynamic model in $\mathbb{R}^3$, Z. Angew. Math. Phys., 70 (2019), Paper No. 7, 16 pp. doi: 10.1007/s00033-018-1049-z.

[37]

Q. XiaoL. Xiong and H. Zhao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials, J. Funct. Anal., 272 (2017), 166-226.  doi: 10.1016/j.jfa.2016.09.017.

[38]

J. Yang and Q. Ju, Global existence of the three-dimensional viscous quantum magnetohydrodynamic model, Journal of Mathematical Physics, 55 (2014), 081501, 12pp. doi: 10.1063/1.4891492.

show all references

References:
[1]

K. Beauchard and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Rational Mech. Anal., 199 (2011), 177-227.  doi: 10.1007/s00205-010-0321-y.

[2]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamical equations, Nonl. Anal., 72 (2010), 4438-4451.  doi: 10.1016/j.na.2010.02.019.

[3]

R. DuanH. LiuS. Ukai and T. Yang, Optimal $L^p$-$L^q$ convergence rate for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233.  doi: 10.1016/j.jde.2007.03.008.

[4]

R. DuanS. UkaiT. Yang and H. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Math. Models Methods Appl. Sci., 17 (2007), 737-758.  doi: 10.1142/S021820250700208X.

[5]

J. Gao, Q. Tao and Z. Yao, Optimal decay rates of classical solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 67 (2016), Art. 23, 22 pp. doi: 10.1007/s00033-016-0616-4.

[6]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Commun. Part. Diff. Eq., 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.

[7]

F. Haas, A magnetohydrodynamic model for quantum plasmas, Phys. Plasmas, 12 (2005), 062117. doi: 10.1063/1.1939947.

[8]

F. Haas, Quantum Plasmas: An Hydrodynamic Approach, Springer, New York, 2011. doi: 10.1007/978-1-4419-8201-8.

[9]

X. Hu and D. Wang, Global solutions to the three-dimisional full compressible magnetohydroynamics flows, Comm. Math. Phys., 283 (2008), 253-284.  doi: 10.1007/s00220-008-0497-2.

[10]

X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.

[11]

N. Jiang and L. Xiong, Diffusive limit of the Boltzmann equation with fluid initial layer in the periodic domain, SIAM J. Math. Anal., 47 (2015), 1747-1777.  doi: 10.1137/130922239.

[12]

T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in the three-dimensinal exterior domain, J. Differ. Equ., 184 (2002), 587-619.  doi: 10.1006/jdeq.2002.4158.

[13]

Y. Kagei and T. Kobayashi, On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in $\Bbb R^3$, Arch. Rational Mech. Anal., 165 (2002), 89-159.  doi: 10.1007/s00205-002-0221-x.

[14]

Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Rational Mech. Anal., 177 (2005), 231-330.  doi: 10.1007/s00205-005-0365-6.

[15]

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 $\Bbb R^3$, Comm. Math. Phys., 200 (1999), 621-659.  doi: 10.1007/s002200050543.

[16]

Y. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232.  doi: 10.1016/j.jmaa.2011.11.006.

[17]

H. Liu and X. Pu, Long Wavelength limit for the quantum Euler-Poisson equation, SIAM J. Math. Anal., 48 (2016), 2345-2381.  doi: 10.1137/15M1046587.

[18]

F. Li and H. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 109-126.  doi: 10.1017/S0308210509001632.

[19]

Q. Liu and C. Zhu, Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations, Indiana Univ. Math. J., 63 (2014), 1085-1108.  doi: 10.1512/iumj.2014.63.5283.

[20]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A., 55 (1979), 337-342.  doi: 10.3792/pjaa.55.337.

[21]

A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464.  doi: 10.1007/BF01214738.

[22]

G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 339-418.  doi: 10.1016/0362-546X(85)90001-X.

[23]

X. Pu and B. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538.  doi: 10.1007/s00033-012-0245-5.

[24]

X. Pu and B. Guo, Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction, Kinet. Related Models, 9 (2016), 165-191.  doi: 10.3934/krm.2016.9.165.

[25]

X. Pu and X. Xu, Decay rates of the magnetohydrodynamic model for quantum plasmas, Z. Angew. Math. Phys., 68 (2017), Paper No. 18, 17 pp. doi: 10.1007/s00033-016-0762-8.

[26]

Z. Tan and H. Wang, Optimal decay rates of the compressible magnetohydrodynamic equations, Nonlinear Analysis: Real World Applications, 14 (2013), 188-201.  doi: 10.1016/j.nonrwa.2012.05.012.

[27]

Z. TanX. Zhang and H. Wang, Asymptotic behavior of Navier-Stokes-Korteweg with friction in $\mathbb{R}^{3}$, Discrete and Continuous Dynamical Systems, 34 (2014), 2243-2259.  doi: 10.3934/dcds.2014.34.2243.

[28]

T. UmedaS. Kawashiwa and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan. J. Appl. Math., 1 (1984), 435-457.  doi: 10.1007/BF03167068.

[29]

S. UkaiT. Yang and H. Zhao, Convergence rate for the compressible Navier-Stokes equations with external force, J. Hyperbolic Differ. Equ., 3 (2006), 561-574.  doi: 10.1142/S0219891606000902.

[30]

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.  doi: 10.1103/PhysRev.40.749.

[31]

Y. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297.  doi: 10.1016/j.jde.2012.03.006.

[32]

J. Wang, Optimal convergence rates for the strong solutions to the compressible Navier-Stokes equations with potential force, Nonlinear Anal. Real World Appl., 34 (2017), 363-378.  doi: 10.1016/j.nonrwa.2016.09.005.

[33]

Y. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271.  doi: 10.1016/j.jmaa.2011.01.006.

[34]

L. WangQ. XiaoL. Xiong and H. Zhao, The Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1049-1097.  doi: 10.1016/S0252-9602(16)30057-1.

[35]

L. Xiong, Incompressible limit of isentropic Navier-Stokes equations with Navier-slip boundary, Kinet. Relat. Models, 11 (2018), 469-490.  doi: 10.3934/krm.2018021.

[36]

X. Xi, X. Pu and B. Guo, Long-time behavior of solutions for the compressible quantum magnetohydrodynamic model in $\mathbb{R}^3$, Z. Angew. Math. Phys., 70 (2019), Paper No. 7, 16 pp. doi: 10.1007/s00033-018-1049-z.

[37]

Q. XiaoL. Xiong and H. Zhao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials, J. Funct. Anal., 272 (2017), 166-226.  doi: 10.1016/j.jfa.2016.09.017.

[38]

J. Yang and Q. Ju, Global existence of the three-dimensional viscous quantum magnetohydrodynamic model, Journal of Mathematical Physics, 55 (2014), 081501, 12pp. doi: 10.1063/1.4891492.

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