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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 and 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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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]

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.

[25]

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

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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]

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.

[25]

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

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