Large-time behavior of entropy solutions to bipolar hydrodynamic model for semiconductors

Abstract

We study the existence and large-time behavior of globally defined entropy solutions to one dimensional bipolar hydrodynamic model for semiconductors on bounded interval. By taking advantage of vanishing viscosity method and compensated compactness framework, we construct approximate solutions uniformly bounded in both space x and time t. Compared with the paper, Huang and Li (2009), a restrictive assumption on the uniform bound with respect to time t of entropy solutions is removed on bounded interval by introducing modified Riemann invariants and invariant region theory, which is a main novelty of this paper. Also, we extend the result for γ∈(1,3] in Huang et al. (2018) to the whole range of physical adiabatic exponent γ∈(1,∞) by using an elementary inequality. Furthermore, we extend the large-time behavior of weak solutions in Huang and Li (2009) from zero doping profile to non-flat one. It is shown that the entropy solution converges to the stationary solution exponentially in time under appropriate condition on doping profile. Some sharp energy estimates are derived to overcome the difficulties due to the coupling and cancellation effect between the difference of densities.