Diffusion relaxation limit of a bipolar hydrodynamic model for semiconductors

In the paper, we discuss the relaxation limit of a bipolar isentropic hydrodynamical models for semiconductors with small momentum relaxation time. With the help of the Maxwell iteration, we prove that, as the relaxation time tends to zero, periodic initial-value problems of a scaled bipolar isentropic hydrodynamic model have unique smooth solutions existing in the time interval where the classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the corresponding drift-diffusion model from the bipolar hydrodynamic model.     

The Asymptotic Behavior and the Quasineutral Limit forthe Bipolar Euler-Poisson System withBoundary Effects and a Vacuum

In this paper, a one-dimensional bipolar Euler-Poisson system(a hydrodynamic model) from semiconductors or plasmas with boundary efects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy’s law time asymptotically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L∞entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the corresponding one-dimensional compressible Euler equations with damping exponentially fast as t → +∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary efects and a vacuum.     

A note on asymptotic behavior of solutions for the one-dimensional bipolar Euler-Poisson system

In this note, we consider a one-dimensional bipolar Euler-Poisson system (hydrodynamic model). This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equations. When n₊≠n₋, paper [I. Gasser, L. Hsiao, H.-L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations 192 (2003) 326-359] discussed the asymptotic behavior of small smooth solutions to the Cauchy problem of the one-dimensional bipolar Euler-Poisson system. Subsequent to [I. Gasser, L. Hsiao, H.-L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations 192 (2003) 326-359], we investigate the asymptotic behavior of solutions to the Cauchy problem with , and obtain the optimal convergence rate toward the constant state . We accomplish the proofs by energy estimates and the decay rates of fundamental solutions of the heat-type equations.     

Relaxation limit and initial layer to hydrodynamic models for semiconductors

We study a relaxation limit of a solution to the initial-boundary value problem for a hydrodynamic model to a drift-diffusion model over a one-dimensional bounded domain. It is shown that the solution for the hydrodynamic model converges to that for the drift-diffusion model globally in time as a physical parameter, called a relaxation time, tends to zero. It is also shown that the solutions to the both models converge to the corresponding stationary solutions as time tends to infinity, respectively. Here, the initial data of electron density for the hydrodynamic model can be taken arbitrarily large in the suitable Sobolev space provided that the relaxation time is sufficiently small because the drift-diffusion model is a coupled system of a uniformly parabolic equation and the Poisson equation. Since the initial data for the hydrodynamic model is not necessarily in “momentum equilibrium”, an initial layer should occur. However, it is shown that the layer decays exponentially fast as a time variable tends to infinity and/or the relaxation time tends to zero. These results are proven by the decay estimates of solutions, which are derived through energy methods.     

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.     

Long-time self-similarity of classical solutions to the bipolar quantum hydrodynamic models

In this paper, a bipolar transient quantum hydrodynamic model (BQHD) for charge density, current density and electric field is considered on the one-dimensional real line. This model takes the form of the classical Euler-Poisson system with additional dispersion caused by the quantum (Bohn) potential. We investigate the long-time behavior of the BQHD model and show the asymptotical self-similarity property of the global smooth solution. Namely, both of the charge densities tend to a nonlinear diffusion wave in large time, which is not a solution to the BQHD equation, but to the combined quasi-neutral, relaxation and semiclassical limiting model. Next, as a by-product, we can compare the large-time behavior of the bipolar quantum hydrodynamic models and of the corresponding classical bipolar hydrodynamic models. As far as we know, the nonlinear diffusion phenomena about the 1D BQHD is new.     

Critical thresholds in hyperbolic relaxation systems

Critical threshold phenomena in one-dimensional 2×2 quasi-linear hyperbolic relaxation systems are investigated. Assuming both the subcharacteristic condition and genuine nonlinearity of the flux, we prove global in time regularity and finite-time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying relaxation systems. Our results apply to the well-known isentropic Euler system with damping. Within the same framework it is also shown that the solution of the semi-linear relaxation system remains smooth for all time, provided the subcharacteristic condition is satisfied.     

一维双极量子漂移-扩散稳态模型的弱解

本文研究了半导体中一维双极量子漂移-扩散稳态模型的弱解.利用指数变换法把此模型转化成两个四阶椭圆方程,然后利用Schauder不动点定理证明了转化后的方程组弱解的存在性.另外得到了方程组解的唯一性和半古典极限.     

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.     

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.     

Relaxation approximation of the Euler equations

The aim of this paper is to show how solutions to the one-dimensional compressible Euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term. The enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate Riemann solver. From a theoretical point of view, the convergence of solutions to the enlarged system towards solutions to the Euler equations is proved for local in time smooth solutions. We also show that arbitrarily large shock waves for the Euler equations admit smooth shock profiles for the enlarged relaxation system. In the end, we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system. We test it on various cases.     

Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors

The global in-time semiclassical and relaxation limits of the bipolar quantum hydrodynamic model for semiconductors are investigated in R³. We prove that the unique strong solution exists and converges globally in time to the strong solution of classical bipolar hydrodynamical equation in the process of semiclassical limit and that of the classical drift-diffusion system under the combined relaxation and semiclassical limits.     

Stability of Non-constant Equilibrium Solutions for Bipolar Full Compressible Navier–Stokes–Maxwell Systems

We study the stability of smooth solutions near non-constant equilibrium states for a bipolar full compressible Navier–Stokes–Maxwell system in a three-dimensional torus \(\mathbb {T}= (\mathbb {R}/\mathbb {Z})^3\). This system is quasilinear hyperbolic-parabolic. In the first part, by using the maximum principle, we find a non-constant steady state solution with small amplitude for this system. In the second part, with the help of suitable choices of symmetrizers and classic energy estimates, we prove that global smooth solutions exist and converge to the non-constant steady states as the time goes to infinity. As a byproduct, we obtain the global stability for the bipolar full compressible Navier–Stokes–Poisson system.     

RELAXATION TIME LIMITS PROBLEM FOR HYDRODYNAMIC MODELS IN SEMICONDUCTOR SCIENCE

In this article, two relaxation time limits, namely, the momentum relaxation time limit and the energy relaxation time limit are considered. By the compactness argument, it is obtained that the smooth solutions of the multidimensional nonisentropic Euler-Poisson problem converge to the solutions of an energy transport model or a drift diffusion model, respectively, with respect to different time scales.     

Global existence versus blow‐up results for one dimensional compressible Navier–Stokes equations with Maxwell's law

We consider one dimensional isentropic compressible Navier–Stokes equations with constitutive relation of Maxwell's law instead of Newtonion law. For this new model, we show that for small initial data, a unique smooth solution exists globally and converges to the equilibrium state as time goes to infinity. For some large data, in contrast to the situation for classical compressible Navier–Stokes equations, which admits global solutions, we show finite time blow up of solutions for the relaxed system. Moreover, we prove the compatibility of the two systems in the sense that, for vanishing relaxation parameters, the solutions to the relaxed system are shown to converge to the solutions of classical system.     

The diffusive relaxation limit of non-isentropic Euler-Maxwell equations for plasmas

This paper concerns the non-isentropic Euler-Maxwell equations for plasmas with short momentum relaxation time. With the help of the Maxwell-type iteration, it is obtained that, as the relaxation time tends to zero, periodic initial-value problem of certain scaled non-isentropic Euler-Maxwell equations has unique smooth solutions existing in the time interval where the corresponding classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the corresponding drift-diffusion model from the non-isentropic Euler-Maxwell equations.     

Numerical treatment of the spherically symmetric solutions of a generalized Fisher-Kolmogorov-Petrovsky-Piscounov equation

In the present work, the connection of the generalized Fisher-KPP equation to physical and biological fields is noted. Radially symmetric solutions to the generalized Fisher-KPP equation are considered, and analytical results for the positivity and asymptotic stability of solutions to the corresponding time-independent elliptic differential equation are quoted. An energy analysis of the generalized theory is carried out with further physical applications in mind, and a numerical method that consistently approximates the energy of the system and its rate of change is presented. The method is thoroughly tested against analytical and numerical results on the classical Fisher-KPP equation, the Heaviside equation, and the generalized Fisher-KPP equation with logistic nonlinearity and Heaviside initial profile, obtaining as a result that our method is highly stable and accurate, even in the presence of discontinuities. As an application, we establish numerically that, under the presence of suitable initial conditions, there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value. An analytical prediction is provided for the Heaviside equation, against which we verify the validity of our computational code, and numerical approximations are provided for several generalized Fisher-KPP problems.     

Global existence and asymptotic behavior of smooth solutions to a bipolar Euler–Poisson equation in a bound domain

In this paper, we present a bipolar hydrodynamic model from semiconductor devices and plasmas, which takes the form of bipolar isentropic Euler–Poisson with electric field and frictional damping added to the momentum equations. We firstly prove the existence of the stationary solutions. Next, we present the global existence and the asymptotic behavior of smooth solutions to the initial boundary value problem for a one-dimensional case in a bounded domain. The result is shown by an elementary energy method. Compared with the corresponding initial data case, we find that the asymptotic state is the stationary solution.     

Long-time behavior of solutions to a nonlinear hyperbolic relaxation system

We study the large time behavior of solutions of a one-dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation. First, we prove the global existence of a unique solution and their decay properties for sufficiently small initial data. We also show that for some large initial data, solutions blow-up in finite time. For quadratic nonlinearities, we prove that the large time behavior of solutions is given by the fundamental solution of the viscous Burgers equation. In some other cases, the convection term is too weak and the large time behavior is given by the linear heat kernel.     

On steady state Euler-Poisson models for semiconductors

This paper is concerned with an analysis of the Euler-Poisson model for unipolar semiconductor devices in the steady state isentropic case. In the two-dimensional case we prove the existence of smooth solutions under a smallness assumption on the prescribed outflow velocity (small boundary current) and, additionally, under a smallness assumption on the gradient of the velocity relaxation time. The latter assumption allows a control of the vorticity of the flow and the former guarantees subsonic flow. The main ingredient of the proof is a regularization of the equation for the vorticity.Also, in the irrotational two- and three-dimensional cases we show that the smallness assumption on the outflow velocity can be replaced by a smallness assumption on the (physical) parameter multiplying the drift-term in the velocity equation. Moreover, we show that solutions of the Euler-Poisson system converge to a solution of the drift-diffusion model as this parameter tends to zero.