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.     

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.     

Combined relaxation and non-relativistic limit of non-isentropic Euler–Maxwell equations

This paper is devoted to study the combined relaxation and non-relativistic limit of non-isentropic Euler–Maxwell equations with relaxation for semiconductors and plasmas. We prove that, as the relaxation time tends to zero and the light speed tends to infinite, periodic initial-value problem of a certain scaled non-isentropic Euler–Maxwell equations has unique smooth solution existing in the time interval where the corresponding classical driftdiffusion model has smooth solutions. It is shown that the relaxation regime plays a decisive role in the combined limit. Furthermore, the corresponding convergence rate is obtained.     

Global convergence of the Euler‐Poisson system for ion dynamics

We consider smooth solutions of the Euler‐Poisson system for ion dynamics in which the electron density is replaced by a Boltzmann relation. The system arises in the modeling of plasmas, where appear two small parameters, the relaxation time and the Debye length. When the initial data are sufficiently close to constant equilibrium states, we prove the convergence of the system for all time, as each of the parameters goes to zero. The limit systems are drift‐diffusion equations and compressible Euler equations. The proof is based on uniform energy estimates and compactness arguments.     

Relaxation-time limit of the multidimensional bipolar hydrodynamic model in Besov space

In this paper, we study a multidimensional bipolar hydrodynamic model for semiconductors or plasmas. This system takes the form of the bipolar Euler-Poisson model with electric field and frictional damping added to the momentum equations. In the framework of the Besov space theory, we establish the global existence of smooth solutions for Cauchy problems when the initial data are sufficiently close to the constant equilibrium. Next, based on the special structure of the nonlinear system, we also show the uniform estimate of solutions with respect to the relaxation time by the high- and low-frequency decomposition methods. Finally we discuss the relaxation-time limit by compact arguments. That is, it is shown that the scaled classical solution strongly converges towards that of the corresponding bipolar drift-diffusion model, as the relaxation time tends to zero.     

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.     

Phase-lag heat conduction: decay rates for limit problems and well-posedness

In two recent papers, the authors have studied conditions on the relaxation parameters in order to guarantee the stability or instability of solutions for the Taylor approximations to dual-phase-lag and three-phase-lag heat conduction equations. However, for several limit cases relating to the parameters, the kind of stability was unclear. Here, we analyze these limit cases and clarify whether we can expect exponential or slow decay for the solutions. Moreover, rather general well-posedness results for three-phase-lag models are presented. Finally, the exponential stability expected by spectral analysis is rigorously proved exemplarily.     

The relaxation-time limit in the compressible Euler–Maxwell equations

The aim of this paper is to study multidimensional Euler–Maxwell equations for plasmas with short momentum relaxation time. The convergence for the smooth solutions to the compressible Euler–Maxwell equations toward the solutions to the smooth solutions to the drift–diffusion equations is proved by means of the Maxwell iteration, as the relaxation time tends to zero. Meanwhile, the formal derivation of the latter from the former is justified.     

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.     

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.     

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.     

On the Cauchy problem for macroscopic model of pedestrian flows

In this paper, we discuss the limit behavior of hyperbolic systems of conservation laws with stiff relaxation terms to the local systems as the relaxation time tends to zero. The prototype is crowd models derived from crowd dynamics according to macroscopic scaling when the flow of crowds is supposed to satisfy the paradigms of continuum mechanics. Under an appropriate structural stability condition, the asymptotic expansion is obtained when one assumes the existence of a smooth solution to the equilibrium system. In this case, the local existence of a classical solution is also shown.     

一类单极等熵流体动力学半导体模型的松弛极限

对一类有短的动量松弛时间的多维等熵流体动力学半导体模型的极限问题进行了讨论．首先构造非线性问题的有初始层的近似解,进而,在归结问题的解存在且有合适的正则性的假设下,证明了原非线性问题的局部古典解的存在性,并且证明了这个解在归结问题解的存在时间区间内收敛到形式近似解．     

Ginzburg‐Landau vortex dynamics driven by an applied boundary current

In this paper we study the time‐dependent Ginzburg‐Landau equations on a smooth, bounded domain Ω ? ?², subject to an electrical current applied on the boundary. The dynamics with an applied current are nondissipative, but via the identification of a special structure in an interaction energy, we are able to derive a precise upper bound for the energy growth. We then turn to the study of the dynamics of the vortices of the solutions in the limit ε → 0. We first consider the original time scale in which the vortices do not move and the solutions undergo a “phase relaxation.” Then we study an accelerated time scale in which the vortices move according to a derived dynamical law. In the dynamical law, we identify a novel Lorentz force term induced by the applied boundary current. © 2010 Wiley Periodicals, Inc.     

SOME RESULTS ON HYPERBOLIC SYSTEMS WITH RELAXATION

In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they prove that the global smooth solutions of the hyperbolic conservation laws systems with relaxation converge to rarefaction waves solution at a determined L^P（p ≥ 2） decay rate.     

First-order L1-convergence for relaxation approximations to conservation laws

We derive a first-order rate of L¹-convergence for stiff relaxation approximations to its equilibrium solutions, i.e., piecewise smooth entropy solutions with finitely many discontinuities for scalar, convex conservation laws. The piecewise smooth solutions include initial central rarefaction waves, initial shocks, possible spontaneous formation of shocks in a future time, and interactions of all these patterns. A rigorous analysis shows that the relaxation approximations to approach the piecewise smooth entropy solutions have L¹-error bound of O(ε|log ε| + ε), where ε is the stiff relaxation coefficient. The first-order L¹-convergence rate is an improvement on the error bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to O(ε). © 1998 John Wiley & Sons, Inc.     

Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations

In this paper, we study the incompressible limit of the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the convergence-stability principle, we show that, when the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient are sufficiently small, the initial-value problem of the model has a unique smooth solution in the time interval where the ideal incompressible magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient go to zero. Moreover, we obtain the convergence of smooth solutions for the model forwards those for the ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.     

The strong relaxation limit of the multidimensional Euler equations

This paper is devoted to the analysis of global smooth solutions to the multidimensional isentropic Euler equations with stiff relaxation. We show that the asymptotic behavior of the global smooth solution is governed by the porous media equation as the relaxation time tends to zero. The results are proved by combining some classical energy estimates with the so-called Shizuta–Kawashima condition.     

The strong relaxation limit of the multidimensional isothermal Euler equations

We construct global smooth solutions to the multidimensional isothermal Euler equations with a strong relaxation. When the relaxation time tends to zero, we show that the density converges towards the solution to the heat equation.

     

A Hierarchy of Models for Two-Phase Flows

Summary. We derive a hierarchy of models for gas-liquid two-phase flows in the limit of infinite density ratio, when the liquid is assumed to be incompressible. The starting model is a system of nonconservative conservation laws with relaxation. At first order in the density ratio, we get a simplified system with viscosity, while at the limit we obtain a system of two conservation laws, the system of pressureless gases with constraint and undetermined pressure. Formal properties of this constraint model are provided, and sticky blocks solutions are introduced. We propose numerical methods for this last model, and the results are compared with the two previous models. Received April 20, 2000; accepted September 12, 2000 %%Online publication November 15, 2000 Communicated by Gérard Iooss