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  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 & 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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

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[15]

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[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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