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.     

QUASI-NEUTRAL LIMIT OF THE BIPOLAR NAVIER-STOKES-POISSON SYSTEM

This paper is concerned with the quasi-neutral limit of the bipolar NavierStokes-Poisson system. It is rigorously proved, by introducing the new modulated energy functional and using the refined energy analysis, that the strong solutions of the bipolar Navier-Stokes-Poisson system converge to the strong solution of the compressible NavierStokes equations as the Debye length goes to zero. Moreover, if we let the viscous coefficients and the Debye length go to zero simultaneously, then we obtain the convergence of the strong solutions of bipolar Navier-Stokes-Poisson system to the strong solution of the compressible Euler equations.     

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.     

The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

The relaxation-time limit from the quantum hydrodynamic model to the quantum drift-diffusion equations in R³ is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift-diffusion model in R³ close to the steady-state is obtained.     

The stationary solution of a one-dimensional bipolar quantum hydrodynamic model

In this paper, we consider the existence and uniqueness of stationary solution to the bipolar quantum hydrodynamic model in one dimensional space with general non-constant doping profile. The existence of the stationary solution is proved by Leray-Schauder fixed-point theorem and a crucial truncation technique is used to derive the positive upper and lower bounds of the stationary solution. The uniqueness of the stationary solution is shown by a delicate energy estimate.     

Zero Mach number limit to compressible quantum magnetohydrodynamic equations with vanishing viscosity coefficients

The connection between the compressible viscous quantum magnetohydrodynamic model with low Mach number and the ideal incompressible magnetohydrodynamic equations is studied in a periodic domain. More precisely, for well‐prepared initial data, we prove the convergence of classical solutions of the compressible viscous quantum magnetohydrodynamic model to the classical solutions of the incompressible ideal magnetohydrodynamic equations with a convergence rate when the Mach number, viscosity coefficient, and magnetic diffusion coefficient simultaneously tend to zero.     

Vanishing viscosity and Debye-length limit to rarefaction wave with vacuum for the 1D bipolar Navier–Stokes–Poisson equation

In this paper, we consider the one-dimensional (1D) compressible bipolar Navier–Stokes–Poisson equations. We know that when the viscosity coefficient and Debye length are zero in the compressible bipolar Navier–Stokes–Poisson equations, we have the compressible Euler equations. Under the case that the compressible Euler equations have a rarefaction wave with one-side vacuum state, we can construct a sequence of the approximation solution to the one-dimensional bipolar Navier–Stokes–Poisson equations with well-prepared initial data, which converges to the above rarefaction wave with vacuum as the viscosity and the Debye length tend to zero. Moreover, we also obtain the uniform convergence rate. The results are proved by a scaling argument and elaborate energy estimate.     

On the blowing up of solutions to one-dimensional quantum Navier-Stokes equations

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the one-dimensional quantum Navier-Stokes equations in a bounded domain is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differen- tial operator, with the quantum Bohm potential, and a density-dependent viscosity. It is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time, if the viscosity constant is not bigger than the scaled Planck constant. The proof is inspired by an observable constructed by Gamba, Gualdani and Zhang, which has been used to study the blowing up of solutions to quantum hydrodynamic models.     

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.     

Quasi-neutral limit of the full bipolar Euler-Poisson system

The quasi-neutral limit of the multi-dimensional non-isentropic bipolar Euler-Poisson system is considered in the present paper. It is shown that for well-prepared initial data the smooth solution of the nonisentropic bipolar Euler-Poisson system converges strongly to the compressible non-isentropic Euler equations as the Debye length goes to zero.     

Large time behavior of solutions to 1-dimensional bipolar quantum hydrodynamic model for semiconductors

In this article, we study the 1-dimensional bipolar quantum hydrodynamic model for semiconductors in the form of Euler-Poisson equations, which contains dispersive terms with third order derivations. We deal with this kind of model in one dimensional case for general perturbations by constructing some correction functions to delete the gaps between the original solutions and the diffusion waves in L²-space, and by using a key inequality we prove the stability of diffusion waves. As the same time, the convergence rates are also obtained.     

LARGE-TIME BEHAVIOR OF SOLUTIONS OF QUANTUM HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

A one-dimensional quantum hydrodynamic model (or quantum Euler-Poisson system) for semiconductors with initial boundary conditions is considered for general pressure-density function. The existence and uniqueness of the classical solution of the corresponding steady-state quantum hydrodynamic equations is proved. Furthermore, the global existence of classical solution, when the initial datum is a perturbation of the steady-state solution, is obtained. This solution tends to the corresponding steady-state solution exponentially fast as the time tends to infinity.     

The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data

The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general (ill-prepared) initial data is rigorously proved in this paper. It is proved that, as the Debye length tends to zero, the solution of the compressible Navier-Stokes-Poisson system converges strongly to the strong solution of the incompressible Navier-Stokes equations plus a term of fast singular oscillating gradient vector fields. Moreover, if the Debye length, the viscosity coefficients and the heat conductivity coefficient independently go to zero, we obtain the incompressible Euler equations. In both cases the convergence rates are obtained.     

A note on barotropic compressible quantum Navier-Stokes equations

The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations has been proved very recently, by Jüngel (2009) [1], if the viscosity constant is smaller than the scaled Plank constant. This paper extends the results to the case that the viscosity constant equals the scaled Plank constant. By using a new estimate on the square root of the solution, apparently not available in [1], the semiclassical limit for the viscous quantum Euler equations (which are equivalent to the barotropic compressible quantum Navier-Stokes equations) can be performed; then the results of this paper are obtained easily.     

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.     

On the blowing up of solutions to quantum hydrodynamic models on bounded domains

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the multi-dimensional quantum hydrodynamic model in a bounded domain is proved. The model consists on conservation of mass equation and a momentum balance equation equivalent to a compressible Euler equations corrected by a dispersion term of the third order in the momentum balance. The proof is based on a priori estimates for the energy functional for a new observable constructed with an auxiliary function, and it is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time. I.M. Gamba is supported by NSF-DMS0507038. M.P. Gualdani acknowledges partial support from the Deutsche Forschungsgemeinschaft, grants JU359/5 and was partially supported under the Feodor Lynen Research fellowship. P. Zhang is partially supported by the NSF of China under Grant 10525101 and 10421101, and the innovation grant from the Chinese Academy of Sciences. Part of the work was done when P. Zhang visited the Department of Mathematics of Texas University at Austin, the author would like to thank the hospitality of the department. Support from the Institute for Computational Engineering and Sciences at the University of Texas at Austin is also gratefully acknowledged.     

The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors

The combined relaxation and vanishing Debye length limit for the hydrodynamic model for semiconductors is considered in both the unipolar and the bipolar case. The resulting limit problems are non‐linear drift driven hyperbolic equations. We make use of non‐standard entropy functions and the related entropy productions in order to obtain uniform estimates. In the bipolar case additional time‐dependent L^∞‐type estimates, available from the existence theory, are needed in order to control the entropy production terms. Finally, strong convergence of the electric field allows the limit towards the limiting problem. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.     

The combined quasineutral and low Mach number limit of Euler–Poisson equations coupled to a magnetic field

In this paper, we consider the combined quasineutral and low Mach number limit of compressible Euler–Poisson system coupled to a magnetic field. We prove that, as the Debye length and the Mach number tend to zero simultaneously in some way, the solution of compressible Euler–Poisson system coupled to a magnetic field will converge to that of ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.