American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry
  • KRM Home
  • This Issue
  • Next Article
    A kinetic model for the formation of swarms with nonlinear interactions
March  2016, 9(1): 165-191. doi: 10.3934/krm.2016.9.165

Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction

Xueke Pu 1, and  Boling Guo 2,

1. 

Department of Mathematics, Chongqing University, Chongqing 401331, China
2. 

Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088

Received  June 2015 Revised  September 2015 Published  October 2015

The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.
Keywords: global solutions, Quantum hydrodynamics, semiclassical limit..
Mathematics Subject Classification: Primary: 35M20; Secondary: 35Q3.
Citation: Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic & Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165
References:
[1]

M. G. Ancona and G. J. Iafrate, Quantum correction to the equation of state of an electron gas in semiconductor, Phys. Rev. B, 39 (1989), 9536-9540. doi: 10.1103/PhysRevB.39.9536.  Google Scholar

[2]

M. G. Ancona and H. F. Tiersten, Macroscopic physics of the silicon inversion layer, Phys. Rev. B, 35 (1987), 7959-7965. doi: 10.1103/PhysRevB.35.7959.  Google Scholar

[3]

D. Bian, L. Yao and C. Zhu, Vanishing capillarity limit of the compressible fluid models of Korteweg type to the Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 1633-1650. doi: 10.1137/130942231.  Google Scholar

[4]

D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" valuables: I; II, Phys. Rev., 85 (1952), 166-179; 180-193. doi: 10.1103/PhysRev.85.166.  Google Scholar

[5]

J. E. Dunn and J. Serrin, On the thermodynamics of interstitial working, Arch. Ration. Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907.  Google Scholar

[6]

R. Feynman, Statistical Mechanics, a Set of Lectures, Reprint of the 1972 original. Advanced Book Classics. Perseus Books, Advanced Book Program, Reading, MA, 1998.  Google Scholar

[7]

C. L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427. doi: 10.1137/S0036139992240425.  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]

H. Hattori and D. Li, Solutions for two-dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X.  Google Scholar

[10]

H. Hattori and D. Li, Global solutions of a high dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069.  Google Scholar

[11]

A. Jungel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068.  Google Scholar

[12]

A. Jungel, C.-K. Lin and K.-C. Wu, An asymptotic limit of a Navier-Stokes system with capillary effects, Comm. Math. Phys., 329 (2014), 725-744. doi: 10.1007/s00220-014-1961-9.  Google Scholar

[13]

A. Jungel and J.-P. Milisic, Full compressible Navier-Stokes equations for quantum fluids: derivation and numerical solution, Kinet. Relat. Models, 4 (2011), 785-807. doi: 10.3934/krm.2011.4.785.  Google Scholar

[14]

L. Hsiao and H. Li, The well-posedness and asymptotics of multi-dimensional quantum hydrodynamics, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 552-568. doi: 10.1016/S0252-9602(09)60053-9.  Google Scholar

[15]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[16]

D. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires par des variations de densité. Arch. Néer. Sci. Exactes Sér, II, 6 (1901), 1-24. Google Scholar

[17]

H. Li and C. K. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors, Comm. Math. Phys., 256 (2005), 195-212. doi: 10.1007/s00220-005-1316-7.  Google Scholar

[18]

H. Li and P. Marcati, Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors, Comm. Math. Phys., 245 (2004), 215-247. doi: 10.1007/s00220-003-1001-7.  Google Scholar

[19]

H. Li and P. Markowich, A review of hydrodynamical models for semiconductors: Asymptotic behavior}, Bol. Soc. Brasil Mat., 32 (2001), 321-342. doi: 10.1007/BF01233670.  Google Scholar

[20]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[21]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  Google Scholar

[22]

X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal., 45 (2013), 834-878. doi: 10.1137/120875648.  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]

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

[25]

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. Google Scholar

show all references

References:
[1]

M. G. Ancona and G. J. Iafrate, Quantum correction to the equation of state of an electron gas in semiconductor, Phys. Rev. B, 39 (1989), 9536-9540. doi: 10.1103/PhysRevB.39.9536.  Google Scholar

[2]

M. G. Ancona and H. F. Tiersten, Macroscopic physics of the silicon inversion layer, Phys. Rev. B, 35 (1987), 7959-7965. doi: 10.1103/PhysRevB.35.7959.  Google Scholar

[3]

D. Bian, L. Yao and C. Zhu, Vanishing capillarity limit of the compressible fluid models of Korteweg type to the Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 1633-1650. doi: 10.1137/130942231.  Google Scholar

[4]

D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" valuables: I; II, Phys. Rev., 85 (1952), 166-179; 180-193. doi: 10.1103/PhysRev.85.166.  Google Scholar

[5]

J. E. Dunn and J. Serrin, On the thermodynamics of interstitial working, Arch. Ration. Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907.  Google Scholar

[6]

R. Feynman, Statistical Mechanics, a Set of Lectures, Reprint of the 1972 original. Advanced Book Classics. Perseus Books, Advanced Book Program, Reading, MA, 1998.  Google Scholar

[7]

C. L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427. doi: 10.1137/S0036139992240425.  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]

H. Hattori and D. Li, Solutions for two-dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X.  Google Scholar

[10]

H. Hattori and D. Li, Global solutions of a high dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069.  Google Scholar

[11]

A. Jungel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068.  Google Scholar

[12]

A. Jungel, C.-K. Lin and K.-C. Wu, An asymptotic limit of a Navier-Stokes system with capillary effects, Comm. Math. Phys., 329 (2014), 725-744. doi: 10.1007/s00220-014-1961-9.  Google Scholar

[13]

A. Jungel and J.-P. Milisic, Full compressible Navier-Stokes equations for quantum fluids: derivation and numerical solution, Kinet. Relat. Models, 4 (2011), 785-807. doi: 10.3934/krm.2011.4.785.  Google Scholar

[14]

L. Hsiao and H. Li, The well-posedness and asymptotics of multi-dimensional quantum hydrodynamics, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 552-568. doi: 10.1016/S0252-9602(09)60053-9.  Google Scholar

[15]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[16]

D. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires par des variations de densité. Arch. Néer. Sci. Exactes Sér, II, 6 (1901), 1-24. Google Scholar

[17]

H. Li and C. K. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors, Comm. Math. Phys., 256 (2005), 195-212. doi: 10.1007/s00220-005-1316-7.  Google Scholar

[18]

H. Li and P. Marcati, Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors, Comm. Math. Phys., 245 (2004), 215-247. doi: 10.1007/s00220-003-1001-7.  Google Scholar

[19]

H. Li and P. Markowich, A review of hydrodynamical models for semiconductors: Asymptotic behavior}, Bol. Soc. Brasil Mat., 32 (2001), 321-342. doi: 10.1007/BF01233670.  Google Scholar

[20]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[21]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  Google Scholar

[22]

X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal., 45 (2013), 834-878. doi: 10.1137/120875648.  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]

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

[25]

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. Google Scholar

[1]

Li Chen, Xiu-Qing Chen, Ansgar Jüngel. Semiclassical limit in a simplified quantum energy-transport model for semiconductors. Kinetic & Related Models, 2011, 4 (4) : 1049-1062. doi: 10.3934/krm.2011.4.1049
[2]

Paolo Antonelli, Pierangelo Marcati. Quantum hydrodynamics with nonlinear interactions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 1-13. doi: 10.3934/dcdss.2016.9.1
[3]

Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669
[4]

Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087
[5]

Harald Friedrich. Semiclassical and large quantum number limits of the Schrödinger equation. Conference Publications, 2003, 2003 (Special) : 288-294. doi: 10.3934/proc.2003.2003.288
[6]

Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599
[7]

José Luis López, Jesús Montejo-Gámez. On viscous quantum hydrodynamics associated with nonlinear Schrödinger-Doebner-Goldin models. Kinetic & Related Models, 2012, 5 (3) : 517-536. doi: 10.3934/krm.2012.5.517
[8]

Jingyu Li, Chuangchuang Liang. Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 833-849. doi: 10.3934/dcds.2016.36.833
[9]

Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic & Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901
[10]

Silvia Cingolani, Mnica Clapp, Simone Secchi. Intertwining semiclassical solutions to a Schrödinger-Newton system. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 891-908. doi: 10.3934/dcdss.2013.6.891
[11]

Christian Rohde, Wenjun Wang, Feng Xie. Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: superposition of rarefaction and contact waves. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2145-2171. doi: 10.3934/cpaa.2013.12.2145
[12]

Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845
[13]

Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic & Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385
[14]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the quantum Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 273-294. doi: 10.3934/cpaa.2017013
[15]

Xuwen Chen, Yan Guo. On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation. Kinetic & Related Models, 2015, 8 (3) : 443-465. doi: 10.3934/krm.2015.8.443
[16]

Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015
[17]

Seok-Jin Kang and Jae-Hoon Kwon. Quantum affine algebras, combinatorics of Young walls, and global bases. Electronic Research Announcements, 2002, 8: 35-46.

[18]

Jihong Zhao, Ting Zhang, Qiao Liu. Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 555-582. doi: 10.3934/dcds.2015.35.555
[19]

Jian Wu, Jiansheng Geng. Almost periodic solutions for a class of semilinear quantum harmonic oscillators. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 997-1015. doi: 10.3934/dcds.2011.31.997
[20]

Peter Markowich, Jesús Sierra. Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic & Related Models, 2019, 12 (2) : 347-356. doi: 10.3934/krm.2019015

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (87)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences