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

TL;DR

This paper proves the global existence and exponential large-time convergence of entropy solutions for a one-dimensional unipolar hydrodynamic model of semiconductors, using vanishing viscosity and invariant region techniques without smallness assumptions.

Contribution

It establishes uniform bounds and convergence results for entropy solutions of the Euler-Poisson system without small initial data constraints.

Findings

Solutions are globally bounded in space and time.

Entropy solutions converge exponentially to stationary states.

No smallness condition on initial data or doping profile.

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.