GLOBAL WEAK SOLUTIONS TO ONE-DIMENSIONAL COMPRESSIBLE VISCOUS HYDRODYNAMIC EQUATIONS

doi:10.1016/S0252-9602(17)30023-1

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (3): 573-583.doi: 10.1016/S0252-9602(17)30023-1

• 论文 • 下一篇

GLOBAL WEAK SOLUTIONS TO ONE-DIMENSIONAL COMPRESSIBLE VISCOUS HYDRODYNAMIC EQUATIONS

郭柏灵¹, 席肖玉²

1. Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China;
2. The Graduate School of China Academy of Engineering Physics, P. O. Box 2101 Beijing 100088, China

收稿日期:2016-03-31 出版日期:2017-06-25 发布日期:2017-06-25
作者简介:Boling GUO,E-mail:gbl@iapcm.ac.cn;Xiaoyu XI,E-mail:xixiaoyu1357@126.com
基金资助:
This work was supported by NSF (11271052).

GLOBAL WEAK SOLUTIONS TO ONE-DIMENSIONAL COMPRESSIBLE VISCOUS HYDRODYNAMIC EQUATIONS

Boling GUO¹, Xiaoyu XI²

1. Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China;
2. The Graduate School of China Academy of Engineering Physics, P. O. Box 2101 Beijing 100088, China

Received:2016-03-31 Online:2017-06-25 Published:2017-06-25
Supported by:
This work was supported by NSF (11271052).

摘要/Abstract

摘要： In this article, we are concerned with the global weak solutions to the 1D compressible viscous hydrodynamic equations with dispersion correction δ²ρ((ϕ(ρ))_xxϕ'(ρ))_x with ϕ(ρ)=ρ^α. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial conditions. The diffusion term εu_xx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in[1] (α=1/2) to 0 < α ≤ 1. In addition, we perform the limit ε → 0 with respect to 0 < α ≤ 1/2.

关键词: Viscous hydrodynamic equations, global weak solution, dispersion correction, periodic boundary and initial conditions

Abstract: In this article, we are concerned with the global weak solutions to the 1D compressible viscous hydrodynamic equations with dispersion correction δ²ρ((ϕ(ρ))_xxϕ'(ρ))_x with ϕ(ρ)=ρ^α. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial conditions. The diffusion term εu_xx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in[1] (α=1/2) to 0 < α ≤ 1. In addition, we perform the limit ε → 0 with respect to 0 < α ≤ 1/2.

Key words: Viscous hydrodynamic equations, global weak solution, dispersion correction, periodic boundary and initial conditions

郭柏灵, 席肖玉. GLOBAL WEAK SOLUTIONS TO ONE-DIMENSIONAL COMPRESSIBLE VISCOUS HYDRODYNAMIC EQUATIONS[J]. 数学物理学报(英文版), 2017, 37(3): 573-583.

Boling GUO, Xiaoyu XI. GLOBAL WEAK SOLUTIONS TO ONE-DIMENSIONAL COMPRESSIBLE VISCOUS HYDRODYNAMIC EQUATIONS[J]. Acta mathematica scientia,Series B, 2017, 37(3): 573-583.

参考文献

[1] Gamba I M, Jüngel A, Vasseur A. Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations. J Differential Equations, 2009, 247(11):3117-3135
[2] Caldeira A, Leggett A. Path integral approach to quantum Brownian motion. Phys A, 1983, 121(3):587-616
[3] Diósi L. On high-temperature Markovian equaton for quantum Brownian motion. Europhys Lett, 1993, 22(1):1-3
[4] Castella F, Erdös L, Frommlet F, Markowich P. Fokker-Planck equations as scaling limits of reversible quantum systems. J Stat Phys, 2000, 100(3/4):543-601
[5] Gualdani M, Jüngel A. Analysis of viscous quantum hydrodynamic equations for semiconductors. European J Appl Math, 2004, 15(5):577-595
[6] Jüngel A. Transport equations for semiconductors. Lecture notes in Phys, vol. 773, Springer, Berlin, 2009
[7] Jüngel A, Milišić J P. Physical and numerical viscosity for quantum hydrodynamics. Commun Math Sci, 2007, 5(2):447-471
[8] Gamba I M, Jüngel A. Positive solutions to singular third order differential equations for quantum fluids. Arch Rational Mech Anal, 2001, 156(3):183-203
[9] Gamba I M, Jüngel A. Asymptotic limits in quantum trajectory models. Comm Patial Differential Equations, 2002, 27(3/4):669-691
[10] Baccarani G, Wordeman M. An investigation of steady-state velocity overshoot in silicon. Solid State Electron, 1985, 28(4):407-416
[11] Antonelli P, Marcati P. On the finite energy weak solutions to a system in quantum fluid dynamics. Comm Math Phys, 2009, 287(2):657-686
[12] Jüngel A. A steady-state quantum Euler-Poisson system for potential flows. Comm Math Phys, 1998, 194(2):463-479
[13] Nishibata S, Suzuki M. Initial boundary value problems for a quantum hydrodynamic model of semiconductors:Asymptotic behaviors and classical limites. J Differential Equations, 2008, 244(4):836-874
[14] Jia Y L, Li H L. Large-time behavior of solutions of quantum hydrodynamic model for semiconductors. Acta Mathematica Scientia, 2006, 26B(1):163-178
[15] Bresch D, Desjardins B, Lin C K. On some compressible fluid models:Korteweg, lubrication, and shallow water systems. Comm Patial Differential Equations, 2003, 28(3/4):843-868
[16] Hsiao L, Li H L. Dissipation and dispersion approximation to hydrodynamic equations and asymptotic limit. J Patial Differential Equations, 2008, 21:59-76
[17] Bresch D, Desjardins B. Quelques modeles diffusifs capillaires de type Korteweg. C R Mecanique, 2004, 332(11):881-886
[18] Lefloch P, Shelukhin V. Symmetries and global solvability of the isothermal gas dynamic equations. Arch Rational Mech Anal, 2005, 175(3):389-430
[19] Chen L, Dreher M. The viscous model of quantum hydrodynamics in several dimensions. Math Models Methods Appl Sci, 2007, 17(7):1065-1093
[20] Gamba I M, Gualdani M, Zhang P. On the blowping up of solutions to the quantum hydrodynamic equations in a bounded domain. Monatsh Math, 2009, 157(1):37-54
[21] Jüngel A. Global weak solutions to compressible Navier-Stokes equations for quantum fluids. Siam J Math Aanl, 2010, 42(3):1025-1045
[22] Li H L, Marcati P. Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors. Comm Math Phys, 2004, 245(2):215-247
[23] Feireisl E. Dynamics of viscous compressible fluids. Oxford University Press, 2004

GLOBAL WEAK SOLUTIONS TO ONE-DIMENSIONAL COMPRESSIBLE VISCOUS HYDRODYNAMIC EQUATIONS

GLOBAL WEAK SOLUTIONS TO ONE-DIMENSIONAL COMPRESSIBLE VISCOUS HYDRODYNAMIC EQUATIONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 3

Metrics

本文评价

推荐阅读 0

[1]	李睿, 赖翀, 吴永洪. GLOBAL WEAK SOLUTIONS TO A GENERALIZED BENJAMIN-BONA-MAHONY-BURGERS EQUATION[J]. 数学物理学报(英文版), 2018, 38(3): 915-925.
[2]	李进开, 辛周平. GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D[J]. 数学物理学报(英文版), 2016, 36(4): 973-1014.
[3]	姚磊; 汪文军. COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY, VACUUM AND GRAVITATIONAL FORCE IN THE CASE OF GENERAL PRESSURE[J]. 数学物理学报(英文版), 2008, 28(4): 801-817.