Uniformly Time-independent L8 Estimate for a One-dimensional Hydrodynamic Model of Semiconductors

It is well known that the a-priori, uniformly time-independent L-infinity 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 alpha(i)(x)not equivalent to 1, in the previous papers, we introduced a technique to obtain the uniform L-infinity 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 alpha(i)(x), and obtain the uniformly, timeindependent estimate of entropy solutions for the physical case alpha i(x) equivalent to 1.