Stability of transonic strong shock waves for the one-dimensional hydrodynamic model for semiconductors
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.
scientific article; zbMATH DE number 2104514
scientific article