Large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductor devices

Abstract

We are concerned with the global existence and large time behavior of entropy solutions to the one dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Poisson equations in a bounded interval. In this paper, we first prove the global existence of entropy solution by vanishing viscosity and compensated compactness framework. In particular, the solutions are uniformly bounded with respect to space and time variables by introducing modified Riemann invariants and the theory of invariant region. Based on the uniform estimates of density, we further show that the entropy solution converges to the corresponding unique stationary solution exponentially in time. No any smallness condition is assumed on the initial data and doping profile. Moreover, the novelty in this paper is about the unform bound with respect to time for the weak solutions of the isentropic Euler-Possion system. 2010 AMS Classification: 35L60, 35L65, 35Q35.

A Sharp Estimate of Entropy Solution to Euler-Poisson System for Semiconductors in the Whole Domain

“…which plays a vital role in deriving the upper bound of density, are not trivial in the whole space. This is very different from the bounded domain case considered in [26]. Fortunately, with the help of relative entropy, we obtain our aim.…”

“…To extend the local solution to the whole space, it is important to obtain a priori estimate of the upper bound of (n ε , J ε ) and the positive lower estimate of n ε . For γ > 1, as stated in [26], [32], the positive lower bound of n ε can be given by the upper bound estimate of J ε n ε . Therefore, we only need to estimate the bounds of Riemann invariants (w ε , z ε ) to gain the upper bound of (n ε , J ε n ε ).…”

“…al. [26] proved the vanishing viscosity weak solution is uniformly bounded by using a maximum principle, and then they proved the weak solution converge to the steady state with an exponential decay rate. However, since the boundedness of the space variable played an essential role in the proof, the method used in [26] can not be used in the whole line.…”

Existence and Large Time Behavior of Entropy Solutions to One-Dimensional Unipolar Hydrodynamic Model for Semiconductor Devices with Variable Coefficient Damping

“…for some positive constant C. Remark 6. Theorems 3 and 5 are generalizations of the corresponding theorem of [18], in which the damping coefficient Hðx, tÞ = −1. Suppose α, β, and λ are three positive constants, then Hðx, tÞ = −α/ð1 + tÞ λ − β satisfies all the assumptions of Hðx, tÞ in Theorems 3 and 5.…”

Global Entropy Solutions to the Gas Flow in General Nozzle

“…To the best of our knowledge, for the isothermal flow, the uniform bound for the approximate solutions depends on time t in all the previous results. We remark that the method in our paper can be applied to obtain the existence of weak solutions of related gas dynamic models, such as Euler-Poisson for a semiconductor model [15] or an Euler equation with geometric source terms [13], and may also shed light on the large time behavior of entropy solutions. Besides, we avoid a laborious numerical scheme to construct approximate solutions.…”