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The asymptotic behavior of global smooth solutions to the hydrodynamic model for semiconductors with spherical symmetry. (English) Zbl 1025.35028
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.

MSC:
35Q60 PDEs in connection with optics and electromagnetic theory
82D37 Statistical mechanical studies of semiconductors
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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