Algebraic time-decay for the bipolar quantum hydrodynamic model

The initial value problem is considered in the present paper for bipolar quantum hydrodynamic model for semiconductors (QHD) in $\mathbb{R}^3$. We prove that the unique strong solution exists globally in time and tends to the asymptotical state with an algebraic rate as $t\to+\infty$. And, we show that the global solution of linearized bipolar QHD system decays in time at an algebraic decay rate from both above and below. This means in general, we can not get exponential time-decay rate for bipolar QHD system, which is different from the case of unipolar QHD model (where global solutions tend to the equilibrium state at an exponential time-decay rate) and is mainly caused by the nonlinear coupling and cancelation between two carriers. Moreover, it is also shown that the nonlinear dispersion does not affect the long time asymptotic behavior, which by product gives rise to the algebraic time-decay rate of the solution of the bipolar hydrodynamical model in the semiclassical limit.Comment: 23 page

Semiclassical and relaxation limits of bipolar quantum hydrodynamic model

The global in-time semiclassical and relaxation limits of the bipolar quantum hydrodynamic model for semiconductors are investigated in $R^3$. We prove that the unique strong solution converges globally in time to the strong solution of classical bipolar hydrodynamical equation in the process of semiclassical limit and to that of the classical Drift-Diffusion system under the combined relaxation and semiclassical limits.Comment: 21 page

Long-time self-similar asymptotic of the macroscopic quantum models

The unipolar and bipolar macroscopic quantum models derived recently for instance in the area of charge transport are considered in spatial one-dimensional whole space in the present paper. These models consist of nonlinear fourth-order parabolic equation for unipolar case or coupled nonlinear fourth-order parabolic system for bipolar case. We show for the first time the self-similarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a self-similar wave (which is not the exact solution of the models) in large time at an algebraic time-decay rate.Comment: 18 page

Global existence and asymptotic behavior of the solutions to the 3D bipolar non-isentropic Euler–Poisson equation

In this paper, the global existence of smooth solutions for the three-dimensional (3D) non-isentropic bipolar hydrodynamic model is showed when the initial data are close to a constant state. This system takes the form of non-isentropic Euler–Poisson with electric field and frictional damping added to the momentum equations. Moreover, the L2-decay rate of the solutions is also obtained. Our approach is based on detailed analysis of the Green function of the linearized system and elaborate energy estimates. To our knowledge, it is the first result about the existence and L2-decay rate of global smooth solutions to the multi-dimensional non-isentropic bipolar hydrodynamic model

Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors

AbstractThe asymptotic behavior of classical solutions of the bipolar hydrodynamical model for semiconductors is considered in the present paper. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equation. The global existence of classical solutions is proven, and the nonlinear diffusive phenomena is observed in large time in the sense that both densities of electron and hole tend to the same unique nonlinear diffusive wave