Uniformly Time-independent L∞ Estimate for a One-dimensional Hydrodynamic Model of Semiconductors

Abstract

It is well known that the a-priori, uniformly time-independent L∞ estimate of solutions is the key point to ensure the large time behavior and the relaxation time limit of entropy solutions for both the unipolar and the bipolar hydrodynamic model of semiconductors. With the technical help of the artificial damping coefficient functions ai(x) ≢ 1, in the previous papers, we introduced a technique to obtain the uniform L∞ estimates of the viscosity-flux approximation solutions, in which, the maximum principle is applied to the combination of the Riemann invariants with the integrals of the density and the concentration of a fixed background charge. In this paper, we remove the auxiliary condition on the functions ai(x), and obtain the uniformly, time-independent estimate of entropy solutions for the physical case ai(x) ≡ 1.