The asymptotic behavior of global smooth solutions to the hydrodynamic model for semiconductors with spherical symmetry
type: Item
instance of: scholarly article
title: The asymptotic behavior of global smooth solutions to the hydrodynamic model for semiconductors with spherical symmetry
zbMATH Open document ID: 1025.35028
DOI: 10.1016/S0362-546X(02)00135-9
author: Ling Hsiao
author: Shu Wang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The paper under review is devoted to the following hydrodynamic model for one carrier transport in semiconductor devices: (1) \(n_t+\nabla\cdot (n{\mathbf u})=0\), (2) \((n{\mathbf u})_t+\nabla\cdot (n{\mathbf u}\otimes {\mathbf u})+\nabla p(n)=n{\mathbf E} - \frac{n{\mathbf u}}{\tau}\) (3) \(\nabla\cdot {\mathbf E} = n-b(x)\) for \((x,t)\in \Omega\times [0,+\infty[\), where \(\Omega=\{x\in\mathbb{R}^d \mid 0<R_1\leq|x|\leq R_2<+\infty\}\). Here \(n\) denotes the electron density, \({\mathbf u}\) the average particle velocity, \({\mathbf E}\) the electric field, \(p=p(n)\) the pressure density function, \(\tau>0\) the momentum relaxation time and \(b=b(x)\) a fixed density. The system (1)-(3) is completed by the following boundary and initial conditions: \[ {\mathbf u} = {\mathbf E} = 0 \text{ on } \partial\Omega \times ]0, + \infty[,\quad n=n_0, \quad {\mathbf u} = {\mathbf u}_0 \text{ on } \Omega\times\{0\}. \] The authors consider spherically symmetric solutions of the form \[ n(x,t)=n(|x|,t), \quad {\mathbf u}=\frac{x}{|x|}u(|x|,t), \quad {\mathbf E}=\frac{x}{|x|} E(|x|,t), \] where \(b(x)=b(|x|)\) in (3). Then (1)-(3) turns into a system for unknown functions defined on \([R_1,R_2]\times [0,+\infty[\). The main results are as follows. 1. Assume \(0 < {\b{b}}\leq b(x)\leq\bar{b}\) for all \(x\in\Omega\). Then there exists a unique solution to the stationary problem. 2. Assume the boundary and initial conditions satisfy an appropriate compatibility condition and a smallness condition including \(\text{osc}_{r\in[R_1,R_2]}b(r)\). Then for sufficiently small \(\tau\), there exists a unique global solution to the nonstationary system. In addition, there holds an asymptotic estimate on the solution. The proof of the first result relies on a fixed point argument, while the proof of the second result is based on a-priori-estimates and the energy method.
reviewed by: Joachim Naumann
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 82D37
Mathematics Subject Classification ID: 76N10
zbMATH DE Number: 1879982
zbMATH Keywords: hydrodynamic model
zbMATH Keywords: semiconductors
zbMATH Keywords: global smooth solution
zbMATH Keywords: asymptotic behavior
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1879982