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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 |
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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