ASYMPTOTIC BEHAVIORS AND CLASSICAL LIMITS OF SOLUTIONS TO A QUANTUM DRIFT-DIFFUSION MODEL FOR SEMICONDUCTORS

2010 ◽  
Vol 20 (06) ◽  
pp. 909-936 ◽  
Author(s):  
SHINYA NISHIBATA ◽  
NAOTAKA SHIGETA ◽  
MASAHIRO SUZUKI
Keyword(s):  
Diffusion Model ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singular Limit ◽  
Drift Diffusion Model ◽  
Planck Constant ◽  
Drift Diffusion ◽  
One Dimensional ◽  
Unique Existence ◽  
Initial Boundary Value

This paper discusses a time global existence, asymptotic behavior and a singular limit of a solution to the initial boundary value problem for a quantum drift-diffusion model of semiconductors over a one-dimensional bounded domain. Firstly, we show a unique existence and an asymptotic stability of a stationary solution for the model. Secondly, it is shown that the time global solution for the quantum drift-diffusion model converges to that for a drift-diffusion model as the scaled Planck constant tends to zero. This singular limit is called a classical limit. Here these theorems allow the initial data to be arbitrarily large in the suitable Sobolev space. We prove them by applying an energy method.

AN INEQUALITY BY GIANAZZA, SAVARÉ AND TOSCANI AND ITS APPLICATIONS TO THE VISCOUS QUANTUM EULER MODEL

2013 ◽  
Vol 15 (05) ◽  
pp. 1250067 ◽  
Author(s):  
XIANGSHENG XU
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Fisher Information ◽  
Gradient Flow ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Drift Diffusion ◽  
Euler Model ◽  
Initial Boundary Value

In this paper we present a simplified version of a coercivity inequality due to Gianazza, Savaré, and Toscani [The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal.194 (2009) 133–220]. Then we use the inequality to construct a weak solution to the initial-boundary value problem for the viscous quantum Euler model.

A Mixture Theory for Quasi-One-Dimensional Diffusion in Fiber-Reinforced Composites

1978 ◽  
Vol 100 (1) ◽  
pp. 128-133 ◽  
Author(s):  
A. Maewal ◽  
G. A. Gurtman ◽  
G. A. Hegemier
Keyword(s):  
Mixture Theory ◽  
Moisture Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fiber Direction ◽  
One Dimensional ◽  
Transient Solutions ◽  
Initial Boundary Value ◽  
Direction Model ◽  
Diffusion Problems

A binary mixture theory is developed for heat transfer in unidirectional fibrous composites with periodic, hexagonal microstructure. The case treated concerns a class of problems for which heat conduction occurs primarily in the fiber direction. Model construction is based upon an asymptotic technique wherein the ratio of transverse-to-longitudinal thermal diffusion times is assumed to be small. The resulting theory contains information on the distribution of temperature and heat flux in individual components. Mixture accuracy is estimated by comparing transient solutions of the mixture equations with finite difference solutions of the Diffusion Equation for an initial boundary value problem. Excellent correlation between “exact” and mixture solutions is observed. The construction procedures utilized herein are immediately applicable to other diffusion problems—in particular, moisture diffusion.

scholarly journals An initial-boundary value problem for Boltzmann’s non-stationary nonlinear one-dimensional four-moment system of equations

2020 ◽  
pp. 91-95
Author(s):  
G. Suleimenov
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Moment Equations ◽  
One Dimensional ◽  
The Moment ◽  
Initial Boundary Value ◽  
The Given ◽  
Dynamic Variables ◽  
Microscopic Distribution

In this article, the set of boundary conditions is defined for first and boundary value problems for the second approximation of Boltzmann’s system of one-dimensional nonlinear moment equations and their logic. For the second approximation of Boltzmann’s one-dimensional non-stationary nonlinear moment equations, which satisfies the Maxwell-Auzhan boundary condition, the theorem for the first boundary problem is considered and by proving this theorem, it is proved that there are only solutions to the given problems. It is known that in many problems of gas dynamics there is no need to describe the complete state of the gas by the function of microscopic distribution of molecules. Therefore, it is better to look for an easier way to describe the gas using macroscopic gas – dynamic variables (density, hydrodynamic average velocity, temperature) are determined in this rotations by the moments of the microscopic distribution function of the molecules, the author faced with the problem of analyzing the different moments of the Boltzmann equation. By studying the moment equations, the author obtained some information about the function of the microscopic distribution of molecules and the convergence of the moment method.

scholarly journals On the existence of global solution of the system of equations of liquid movement in porous medium

2021 ◽  
Vol 234 ◽  
pp. 00095
Author(s):  
Margarita Tokareva ◽  
Alexander Papin
Keyword(s):  
Porous Medium ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Solvability ◽  
Compressible Liquid ◽  
Fluid Filtration ◽  
One Dimensional ◽  
Lagrangian Variables ◽  
Initial Boundary Value ◽  
Liquid Movement

The initial-boundary value problem for the system of one-dimensional isothermal motion of viscous liquid in deformable viscous porous medium is considered. Local theorem of existence and uniqueness of problem is proved in case of compressible liquid. In case of incompressible liquid the theorem of global solvability in time is proved in Holder classes. A feature of the model of fluid filtration in a porous medium considered in this paper is the inclusion of the mobility of the solid skeleton and its poroelastiс properties. The transition from Euler variables to Lagrangian variables is used in the proof of the theorems.

scholarly journals Filtration of Liquid in a Non-isothermal Viscous Porous Medium

2020 ◽  
pp. 763-773
Author(s):  
Alexander A. Papin ◽  
Margarita A. Tokareva ◽  
Rudolf A. Virts
Keyword(s):  
Porous Medium ◽  
Boundary Value Problem ◽  
Fluid Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Heat Conducting ◽  
System Of Equations ◽  
One Dimensional ◽  
Unsteady Fluid ◽  
Initial Boundary Value

The solvability of the initial-boundary value problem is proved for the system of equations of one-dimensional unsteady fluid motion in a heat-conducting viscous porous medium

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