Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors

Abstract

In this paper, we study the zero relaxation time limits to a one dimensional hydrodynamic model of two carrier types for semiconductors. First, we introduce the flux approximation coupled with the classical viscosity method to obtain the uniform $$L_{loc}^{p}, p \ge 1, $$ bound of the approximation solutions $$ \rho _{i}^{ \varepsilon ,\delta } $$ and other estimates of $$ (u_{i}^{ \varepsilon ,\delta }, E^{ \varepsilon ,\delta })$$ with the help of the high energy estimates (Jungel and Peng Comm Partial Differ Equ 58:1007–1033, 1999). Then, we apply the compensated compactness method coupled with the scaled variables technique (Marcati and Natalini Arch Ration Mech Anal 129:129–145, 1995) to prove the zero-relaxation-time limits with arbitrarily large initial data, and arbitrary adiabatic exponents $$ \gamma _{i} > 1$$ .