The asymptotic behaviour of global smooth solutions to the multi‐dimensional hydrodynamic model for semiconductors

We establish the global existence of smooth solutions to the Cauchy problem for the multi‐dimensional hydrodynamic model for semiconductors, provided that the initial data are perturbations of a given stationary solutions, and prove that the resulting evolutionary solution converges asymptotically in time to the stationary solution exponentially fast. Copyright © 2003 John Wiley & Sons, Ltd.     

Asymptotic stability of solutions to the nonisentropic hydrodynamic model for semiconductors

In this paper, the asymptotic stability of smooth solutions to the multidimensional nonisentropic hydrodynamic model for semiconductors is established, under the assumption that the initial data are a small perturbation of the stationary solutions for the thermal equilibrium state, whose proofs mainly depend on the basic energy methods.     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Asymptotic Behavior of Global Smooth Solotions to the Euler-Poisson System in Semiconductors

In this paper, we establish the global existence and the asymptotic behavior of smooth solution to the initial-boundary value problem of Euler-Poisson system which is used as the bipolar hydrodynamic model for semiconductors with the nonnegative constant doping profile.     

Global existence and large time behavior of solutions for the bipolar quantum hydrodynamic models in the quarter plane

In this paper, we discuss a bipolar transient quantum hydrodynamic model for charge density, current density, and electric field in the quarter plane. This model takes the form of a classical Euler–Poisson system with the additional dispersion terms caused by the quantum (Bohn) potential. We show global existence of smooth solutions for the initial boundary value problem when the initial data are near the nonlinear diffusive waves, which are different from the steady state. We also show the asymptotical behavior of the global smooth solution towards the nonlinear diffusive waves and obtain the algebraic decay rates. These results are proved by elaborate energy methods. Finally, using the Fourier analysis, we obtain the optimal convergence rates of the solutions towards the nonlinear diffusion waves. As far as we known, this is the first result about the initial boundary value problem of the one‐dimensional bipolar quantum hydrodynamic model in the quarter plane. Copyright © 2013 John Wiley & Sons, Ltd.     

Quasineutral limit of bipolar quantum hydrodynamic model for semiconductors

This paper is concerned with the quasineutral limit of the bipolar quantum hydrodynamic model for semiconductors. It is rigorously proved that the strong solutions of the bipolar quantum hydrodynamic model converge to the strong solution of the so-called quantum hydrodynamic equations as the Debye length goes to zero. Moreover, we obtain the convergence of the strong solutions of bipolar quantum hydrodynamic model to the strong solution of the compressible Euler equations with damping if both the Debye length and the Planck constant go to zero simultaneously.     

Diffusion relaxation limit of a nonisentropic hydrodynamic model for semiconductors

In this paper, we discussed a general multidimensional nonisentropic hydrodynamical model for semiconductors with small momentum relaxation time. The model is self‐consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. With the help of the Maxwell‐type iteration, we prove that, as the relaxation time tends to zero, periodic initial‐value problem of certain scaled multidimensional nonisentropic hydrodynamic model has a unique smooth solution existing in the time interval where the corresponding classical drift‐diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the drift‐diffusion models from the nonisentropic hydrodynamic models. Copyright © 2007 John Wiley & Sons, Ltd.     

Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions

The existence of global smooth solutions to the multidimensional hydrodynamic model for plasmas of electrons and positively charged ions with insulating boundary conditions is shown under the assumption that the initial densities are close to a constant. Furthermore it is proved that the particle densities converge exponentially fast to the constant steady state.     

Quasineutral limit of the viscous quantum hydrodynamic model for semiconductors

The quasineutral limit (zero-Debye-length limit) of viscous quantum hydrodynamic model for semiconductors is studied in this paper. By introducing new modulated energy functional and using refined energy analysis, it is shown that, for well-prepared initial data, the smooth solution of viscous quantum hydrodynamic model converges to the strong solution of incompressible Navier-Stokes equations as the Debye length goes to zero.     

The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors

In this paper we study the asymptotic behavior of globally smooth solutions of the Cauchy problem for the multidimensional isentropic hydrodynamic model for semiconductors in R^d. We prove that smooth solutions (close to equilibrium) of the problem converge to a stationary solution exponentially fast as t→+∞.     

Asymptotic profile in a one-dimensional steady-state nonisentropic hydrodynamic model for semiconductors

We study the stationary flow for a one-dimensional nonisentropic hydrodynamic model for semiconductor devices. This model consists of the continuous equations for the electron density, the electron current density and electron temperature, coupled the Poisson equation of the electrostatic potential. In a bounded interval supplemented by the proper boundary conditions, we investigate the zero-electron-mass limit, the zero-relaxation-time limit and the Debye-length (quasi-neutral) limit, respectively. We show the strong convergence of the sequence of solutions and give the associated convergence rate.     

Existence and stability of steady-state of one-dimensional quantum hydrodynamic system for semiconductors

A transient quantum hydrodynamic system for charge density, current density and electrostatic potential is considered in spatial one-dimensional real line. The equations take the form of classical Euler-Poisson system with additional dispersion caused by the quantum (Bohm) potential and used, for instance, to account for quantum mechanical effects in the modelling of charge transport in ultra submicron semiconductor devices such as resonant tunnelling trough oxides gate and inversion layer energy quantization and so on.The existence and uniqueness and long time stability of steady-state solution with spatial different end states and large strength is proven in Sobolev space. To guarantee the existence and stability, we propose a stability condition which can be viewed as a quantum correction to classical subsonic condition. Furthermore, since the argument for classical hydrodynamic equations does not apply here due to the dispersion term, we also show the local-in-time existence of strong solution in terms of a reformulated system for the charge density and the electric field consisting of two coupled semilinear (spatial) fourth-order wave type equations.     

Asymptotic behaviour of nonlinear difference equations

In this paper, we investigate the growth/decay rate of solutions of a class of nonlinear Volterra difference equations. Our results can be applied for the case when the characteristic equation of an associated linear difference equation has complex dominant eigenvalue with higher than one multiplicity. Illustrative examples are given for describing the asymptotic behaviour of solutions in a class of linear difference equations and in several discrete nonlinear population models.     

Global smooth solution of hydrodynamical equation for the Heisenberg paramagnet

In this note, we obtain the existence and uniqueness of global smooth solution for the Cauchy problem of multidimensional hydrodynamical equation for the Heisenberg paramagnet. Copyright © 2004 John Wiley & Sons, Ltd.     

Asymptotic behaviour of the energy for electromagnetic systems with memory

Some linear evolution problems arising in the theory of hereditary electromagnetism are considered here. Making use of suitable Liapunov functionals, existence of solutions as well as asymptotic behaviour, are determined for rigid conductors with electric memory. In particular, we show the polynomially decay of the solutions, when the memory kernel decays exponentially or polynomially. Copyright © 2004 John Wiley & Sons, Ltd.