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Weak solutions to a hydrodynamic model for semiconductors: The Cauchy problem. (English) Zbl 0831.35157
The paper is concerned with the following system of PDEs (hydrodynamic model for a one-dimensional semiconductor device): \[ (1)\;n_t + (nu)_x = 0, \quad (2)\;(nu)_t + \bigl( nu^2 + p(n) \bigr)_x = nE - nu, \quad (3)\;E_x = n - b \] in the region \(\mathbb{R} \times [0,T)\), where \(n\) denotes the electron density, \(u\) the average particle velocity, \(E\) the (negative) electric field and \(b\) the doping profile; \(p = p(n)\) is a given pressure function. System (1)–(3) is completed by initial conditions on \(n\) and \(u\).
The aim of the paper is to prove the existence of locally bounded weak solutions to (1)–(3), where \(n\), \(Ju = nu\) (= electron current density) and \(E\) are the dependent variables. The authors construct approximate solutions by using two finite difference schemes: the fractional step Lax-Friedrichs scheme and the Godounov scheme. Then supremum-norm bounds and energy-type estimates on the approximate solutions are established. The passage to the limit is then carried out by the aid of compensated compactness techniques.
Reviewer: J.Naumann (Berlin)

MSC:
35Q60 PDEs in connection with optics and electromagnetic theory
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
35D05 Existence of generalized solutions of PDE (MSC2000)
78A35 Motion of charged particles
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