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Abstract：We consider a onedimensional bipolar isentropic quantum hydrodynamical model from semiconductor devices.First，we discuss the classical limit of the stationary solution.Then we discuss the classical limit of the nonstationary initial boundary problem for a onedimensional case in a bounded domain.We show that the solutions to the quantum hydrodynamic model of semiconductors approaches that to the hydrodynamic model of semiconductors as the scaled Planck constants ε tends to zero.
Key words：bipolar； quantum hydrodynamic model； classical limit
CLC number： O 29 Document code： A Article ID： 10005137（2015）02011111
References：
[1] GARDNER C.The quantum hydrodynamic model for semiconductors devices[J].Appl Math，1994，54：409-427.
[2] JNGEL A.Quasihydrodynamic semiconductor equations，Progress in Nonlinear Differential Equations[M].Boston：Birkhuser，2001.
[3] LUO T，NATALINI R，XIN Z.Large time behavior of the solutions to a hydrodynamic model for semiconductors[J].Appl Math，1998，59：810-830.
[4] NISHBATA S，SUZUKI M.Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors[J].Math，2007，44：639-665.
[5] NISHBATA S，SUZUKI M.Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors[J].Arch Ration Mech Anal，2008，244：836-874.
[6] NATALIN Iatalini R.The bipolar hydrodynamic model for semiconductors and the driftdiffusion equation[J].Math Anal Appl，1996，198：262-281.
[7] HSIAO L，ZHANG K J.The relaxation of the hydrodynamic model for semiconductors to the driftdiffusion equations[J].Differential Equations，2000，165：315-354.
[8] HSIAO L，ZHANG K J.The global weak solution and relaxation limits of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors[J].Math Models Methods Appl Sci，2000，10：1333-1361.
[9] ZHU C，HATTORI H.Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species[J].Differential Equations，2000，166：1-32.
[10] LATTANZIO C.On the 3D bipolar isentropic EulerPoisson model for semiconductors and the driftdiffusion limit[J].Math Models Methods Appl Sci，2000，10：351-360.
[11] GASSER I，MARCATI P.The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors[J].Math Meth Appl Sci，2001，24：81-92.
[12] GASSER I，HSIAO L，LI H L.Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors[J].Differential Equations，2003，192：326-359. [13] HUANG F M，LI Y P.Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum[J].Dis Cont Dyn Sys，2009，24：455-470.
[14] ZHANG G J，LI H L，ZHANG K J.Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors[J].Differential Equations，2008，245：1433-1453.
[15] HUANG F，LI H L，MATSUMURA A.Existence and stability of steadystate of onedimensional quantum EulerPoisson system for semiconductors[J].Differential Equations，2006，225：1-25.
[16] LI H L，MARCATI P.Existence and asymptotic behaviour of multidimensional quantum hydrodynamic model for semiconductors[J].Math Phys，2004，245：215-247.
[17] NISHBATA S，SUZUKI M.Initial boundary value problems for a quantum hydrodynamic model of semiconductors：Asymptotic behaviors and classical limits[J].Differential Equations，2008，244：836-874.
[18] TSUGE N.Existence and uniqueness of stationary solutions to a onedimensional bipolar hydrodynamic models of semiconductors[J].Nonlinear Anal，2010，73：779-787.
[19] ZHOU F，LI Y P.Existence and some limits of stationary solutions to a onedimensional bipolar EulerPoisson system[J].Math Anal Appl，2009，351：480-490.
[20] LI Y P.Global existence and asymptotic behavior of smooth to a bipolar EulerPoisson equation in a bound domain[J].Math Phys Springer，2012，10：1125-1144.
摘要：考虑一维双极等熵量子力学模型.首先，对方程进行一些变形，利用Poincarés不等式及函数收敛和弱收敛的一些性质，得到了稳态解的经典极限，即当普朗克常量ε趋于0时，量子力学模型方程的稳态解趋于经典力学模型方程的稳态解.然后，利用非稳态解已有的一些结论和Sobolev不等式，Schwartz不等式，Gronwall不等式及一些能量估计，得到了非稳态解的经典极限，即量子力学模型方程的光滑解趋于经典力学模型方程的光滑解.
关键词：双极； 量子力学模型； 经典极限
（责任编辑：冯珍珍）
Key words：bipolar； quantum hydrodynamic model； classical limit
CLC number： O 29 Document code： A Article ID： 10005137（2015）02011111
References：
[1] GARDNER C.The quantum hydrodynamic model for semiconductors devices[J].Appl Math，1994，54：409-427.
[2] JNGEL A.Quasihydrodynamic semiconductor equations，Progress in Nonlinear Differential Equations[M].Boston：Birkhuser，2001.
[3] LUO T，NATALINI R，XIN Z.Large time behavior of the solutions to a hydrodynamic model for semiconductors[J].Appl Math，1998，59：810-830.
[4] NISHBATA S，SUZUKI M.Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors[J].Math，2007，44：639-665.
[5] NISHBATA S，SUZUKI M.Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors[J].Arch Ration Mech Anal，2008，244：836-874.
[6] NATALIN Iatalini R.The bipolar hydrodynamic model for semiconductors and the driftdiffusion equation[J].Math Anal Appl，1996，198：262-281.
[7] HSIAO L，ZHANG K J.The relaxation of the hydrodynamic model for semiconductors to the driftdiffusion equations[J].Differential Equations，2000，165：315-354.
[8] HSIAO L，ZHANG K J.The global weak solution and relaxation limits of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors[J].Math Models Methods Appl Sci，2000，10：1333-1361.
[9] ZHU C，HATTORI H.Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species[J].Differential Equations，2000，166：1-32.
[10] LATTANZIO C.On the 3D bipolar isentropic EulerPoisson model for semiconductors and the driftdiffusion limit[J].Math Models Methods Appl Sci，2000，10：351-360.
[11] GASSER I，MARCATI P.The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors[J].Math Meth Appl Sci，2001，24：81-92.
[12] GASSER I，HSIAO L，LI H L.Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors[J].Differential Equations，2003，192：326-359. [13] HUANG F M，LI Y P.Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum[J].Dis Cont Dyn Sys，2009，24：455-470.
[14] ZHANG G J，LI H L，ZHANG K J.Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors[J].Differential Equations，2008，245：1433-1453.
[15] HUANG F，LI H L，MATSUMURA A.Existence and stability of steadystate of onedimensional quantum EulerPoisson system for semiconductors[J].Differential Equations，2006，225：1-25.
[16] LI H L，MARCATI P.Existence and asymptotic behaviour of multidimensional quantum hydrodynamic model for semiconductors[J].Math Phys，2004，245：215-247.
[17] NISHBATA S，SUZUKI M.Initial boundary value problems for a quantum hydrodynamic model of semiconductors：Asymptotic behaviors and classical limits[J].Differential Equations，2008，244：836-874.
[18] TSUGE N.Existence and uniqueness of stationary solutions to a onedimensional bipolar hydrodynamic models of semiconductors[J].Nonlinear Anal，2010，73：779-787.
[19] ZHOU F，LI Y P.Existence and some limits of stationary solutions to a onedimensional bipolar EulerPoisson system[J].Math Anal Appl，2009，351：480-490.
[20] LI Y P.Global existence and asymptotic behavior of smooth to a bipolar EulerPoisson equation in a bound domain[J].Math Phys Springer，2012，10：1125-1144.
摘要：考虑一维双极等熵量子力学模型.首先，对方程进行一些变形，利用Poincarés不等式及函数收敛和弱收敛的一些性质，得到了稳态解的经典极限，即当普朗克常量ε趋于0时，量子力学模型方程的稳态解趋于经典力学模型方程的稳态解.然后，利用非稳态解已有的一些结论和Sobolev不等式，Schwartz不等式，Gronwall不等式及一些能量估计，得到了非稳态解的经典极限，即量子力学模型方程的光滑解趋于经典力学模型方程的光滑解.
关键词：双极； 量子力学模型； 经典极限
（责任编辑：冯珍珍）