Stability of transonic strong shock waves for the one-dimensional hydrodynamic model for semiconductors
type: Item
instance of: scholarly article
title: Stability of transonic strong shock waves for the one-dimensional hydrodynamic model for semiconductors
zbMATH Open document ID: 1061.35050
published in: Journal of Differential Equations
publication date: 2004-10-01T00:00:00Z
review text: The Boltzmann equation models the flow of the electrons in semiconductor crystals. The hydrodynamic equations can be derived from the Boltzmann equation by using a moment method. Mathematically, the hydrodynamic model is a system of quasilinear hyperbolic-elliptic equations which represent the fundamental laws of balance of particles, momentum and energy of electrons. Thus, the solution can become discontinuous in finite time, and the occurrence of shocks has to be taken into account. In this paper the author investigates the stability of strong transonic shock waves for the solution of the one-dimensional hydrodynamic model for unipolar semiconductors in a steady state. The analysis of these solutions is based on the construction of their orbits in the phase volume and on the representation of discontinuous solutions by union of trajectory pieces. This approach is performed by Asher et al. studying the phase plane portrait for the limit case of an isentropic flow with an adiabatic exponent \(\gamma\) equal to one, i.e. the isothermal hydrodynamic model. Markowich discussed the more realistic case \(\gamma=5/3\), but under the restrictive assumption of an infinite current relaxation time (electron plasma). Rosini generalized these results by considering an arbitrary, but fixed adiabatic exponent \(\gamma> 1\) and taking also into account scattering events of the electrons. The goal is to analyse the stability of the stationary transonic solutions \(V\) with a singe shock wave front \(\Sigma\), separating two given states \(V^+= (\rho^+, J,E^+)^T\) and \(V^-= (\rho^-, J,E^-)^T\), obeying the Rankine-Hugoniot jump conditions and the Lax entropy conditions.
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 82D37
zbMATH DE Number: 2104514
zbMATH Keywords: hyperbolic-elliptic system
zbMATH Keywords: unipolar semiconductor
zbMATH Keywords: Rankine-Hugoniot jump conditions
zbMATH Keywords: Lax entropy conditions
author: Massimiliano D. Rosini
reviewed by: Stefan Balint
MaRDI profile type: Publication
full work available at URL: j.jde.2003.09.009
OpenAlex ID: W2050230295
cites work: A PHASE PLANE ANALYSIS OF TRANSONIC SOLUTIONS FOR THE HYDRODYNAMIC SEMICONDUCTOR MODEL
cites work: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires
cites work: Q4016476
cites work: Q3947998
cites work: Q5801144
cites work: Stability of Step Shocks
cites work: Numerical approximation of hyperbolic systems of conservation laws
cites work: Q3251949
cites work: Initial boundary value problems for hyperbolic systems
cites work: Hyperbolic systems of conservation laws II
cites work: The stability of multidimensional shock fronts
cites work: The existence of multidimensional shock fronts
cites work: Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation
cites work: Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem
cites work: Q4273686
cites work: Q3808553
cites work: Q2751138
cites work: Q3927555
cites work: Q3435494
DOI: 10.1016/J.JDE.2003.09.009
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description: scientific article; zbMATH DE number 2104514