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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

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.
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. 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. 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. 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. 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. 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, (1972). Google Scholar

[7]

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

[8]

F. Haas, Quantum Plasmas: An Hydrodynamic Approach,, Springer, (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. 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. 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. 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. 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. 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. 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. 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, 6 (1901), 1. 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. 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. 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. 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. Google Scholar

[21]

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

[22]

X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions,, SIAM J. Math. Anal., 45 (2013), 834. 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. 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. doi: 10.1016/j.jmaa.2011.01.006. Google Scholar

[25]

E. Wigner, On the quantum correction for thermodynamic equilibrium,, \emph{Phys. Rev.}, 40 (1932), 749. 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. 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. 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. 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. 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. 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, (1972). Google Scholar

[7]

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

[8]

F. Haas, Quantum Plasmas: An Hydrodynamic Approach,, Springer, (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. 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. 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. 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. 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. 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. 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. 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, 6 (1901), 1. 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. 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. 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. 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. Google Scholar

[21]

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

[22]

X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions,, SIAM J. Math. Anal., 45 (2013), 834. 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. 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. doi: 10.1016/j.jmaa.2011.01.006. Google Scholar

[25]

E. Wigner, On the quantum correction for thermodynamic equilibrium,, \emph{Phys. Rev.}, 40 (1932), 749. Google Scholar

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