Asymptotic Analysis of a Semiconductor Model Based on Fermi–Dirac Statistics

The quasi-hydrodynamic carrier transport equations for semiconductors extended to Fermi–Dirac statistics are considered. It is shown that in the high injection case, these equations reduce to a drift-diffusion model with non-linear diffusion terms. The limiting procedure is proved rigorously and error estimates are shown. We compute numerically static voltage–current characteristics of a forward biased pn-junction diode and compare the curves with the corresponding characteristics obtained from the standard drift-diffusion model based on Boltzmann statistics. It turns out that there exists a so-called threshold voltage at which the behaviour of the characteristic changes. Under high injection conditions, the dependence of the current on the bias appears to be approximately polynomial. The characteristics are studied analytically for a unipolar device.     

Self‐attracting Fermi–Dirac particles in canonical and microcanonical setting

The existence of stationary solutions for the system describing the interaction of gravitationally attracting particles that obey the Fermi–Dirac statistics is studied depending on the relations between mass, the energy and the temperature of the particles. Copyright © 2005 John Wiley & Sons, Ltd.     

Lax方程与耦合Poisson流形的动力r-矩阵

对于一个与Poisson流形耦合的动力r-矩阵,我们在相应的Lie双代数胚上构造出一类Lax方程和一族守恒量,希望利用该方法进一步研究可积Hamilton系统.     

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

We present a rigorous derivation of the Ericksen‐Leslie equation starting from the Doi‐Onsager equation by the Hilbert expansion method. The existence of the Hilbert expansion is related to an open question of whether the energy of the Ericksen‐Leslie equation is dissipated. On this point, we show that the energy is dissipated for the Ericksen‐Leslie equation derived from the Doi‐Onsager equation. The most difficult step is to prove a uniform bound for the remainder of the Hilbert expansion. This step is connected to the spectral stability of the linearized Doi‐Onsager operator around a critical point and the lower bound estimate for a bilinear form associated with the linearized operator. By introducing two important auxiliary operators, we can obtain the detailed spectral information for the linearized operator around all the critical points. We establish a precise lower bound of the bilinear form by introducing a five‐dimensional space called the Maier‐Saupe space.© 2015 Wiley Periodicals, Inc.     

定态耦合Harry-Dym方程的Bargmann坐标及约束系统的Poisson结构

该文首先给出相联于耦合Harry-Dym(CHD)族的Lenard递归方程的多项式解,并证明了任一定态CHD方程的解均有可积的Bargmann坐标表示.最后讨论了约束系统的动力r-矩阵及Poisson结构.     

Homogenization and Enhancement of the G‐Equation in Random Environments

We study the homogenization of a G‐equation that is advected by a divergence free “small mean” stationary vector field in a general ergodic random environment. We prove that the averaged equation is an anisotropic deterministic G‐equation, and we give necessary and sufficient conditions for enhancement. Since the problem is not assumed to be coercive, it is not possible to have uniform bounds for the solutions. In addition, as we show, the associated minimal (first passage) time function does not satisfy, in general, the uniform integrability condition that is necessary to apply the subadditive ergodic theorem. We overcome these obstacles by (i) establishing a new reachability (controllability) estimate for the minimal function and (ii) constructing, for each direction and almost surely, a random sequence that has both a long‐time averaged limit (due to the subadditive ergodic theorem) and stays asymptotically close to the minimal time. © 2013 Wiley Periodicals, Inc.     

Poisson Kernel for a Degenerate Elliptic Equation and Application to Boundary Value Problems

In this paper we consider the Dirichlet problem to a degenerate elliptic equation in a domain whose interior contains a degenerate surface. By means of the method of expansion of Poisson kernel and applying the properties of special functions, we obtain the twice continuously differentiable solution of the problem on the entire space including infinity.     

随机环境中独立过程的Hydrodynamic极限

本文研究随机环境中独立过程的Hydrodynamic极限,在恰当的条件下,构造出了随机环境中的独立过程,并通过对偶公式得到了该粒子模型的宏观偏微分方程。     

A Quasi-Neutral Limit in a Hydrodynamic Model for Charged Fluids

 The combined quasineutral and relaxation time limit for a bipolar hydrodynamic model is considered. The resulting limit problem is a nonlinear diffusion equation describing a neutral fluid. We make use of various entropy functions and the related entropy productions in order to obtain strong enough uniform bounds. The necessary strong convergence of the densities is obtained by using a generalized version of the “div-curl” Lemma and monotonicity methods.     

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.     

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．     

A Quasi-Neutral Limit in a Hydrodynamic Model for Charged Fluids

 The combined quasineutral and relaxation time limit for a bipolar hydrodynamic model is considered. The resulting limit problem is a nonlinear diffusion equation describing a neutral fluid. We make use of various entropy functions and the related entropy productions in order to obtain strong enough uniform bounds. The necessary strong convergence of the densities is obtained by using a generalized version of the “div-curl” Lemma and monotonicity methods. Received September 27, 2001; in revised form February 25, 2002     

On a boundary value problem for a p‐Dirac equation

The p‐Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p‐Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p‐Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p‐Laplace equation into the p‐Dirac equation. This equation will be solved iteratively by using a fixed‐point theorem. Applying operator‐theoretical methods for the p‐Dirac equation and p‐Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.     

Initial‐Boundary Value Problems for the Coupled Nonlinear Schrödinger Equation on the Half‐Line

Initial‐boundary value problems for the coupled nonlinear Schrödinger equation on the half‐line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k‐plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrödinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so‐called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.     

Directional Derivative Problem for the Telegraph Equation with a Dirac Potential

In the domain Q = [0,∞)×[0,∞) of the variables (x, t), for the telegraph equation with a Dirac potential concentrated at a point (x₀, t₀) ∈ Q, we consider a mixed problem with initial (at t = 0) conditions on the solution and its derivative with respect to t and a condition on the boundary x = 0 which is a linear combination with coefficients depending on t of the solution and its first derivatives with respect to x and t (a directional derivative). We obtain formulas for the classical solution of this problem under certain conditions on the point (x₀, t₀), the coefficient of the Dirac potential, and the conditions of consistency of the initial and boundary data and the right-hand side of the equation at the point (0, 0). We study the behavior of the solution as the direction of the directional derivative in the boundary condition tends to a characteristic of the equation and obtain estimates of the difference between the corresponding solutions.     

Boundary Layers and Incompressible Navier‐Stokes‐Fourier Limit of the Boltzmann Equation in Bounded Domain I

We establish the incompressible Navier‐Stokes‐Fourier limit for solutions to the Boltzmann equation with a general cutoff collision kernel in a bounded domain. Appropriately scaled families of DiPerna‐Lions(‐Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to 0. Every limit point is a weak solution to the Navier‐Stokes‐Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of the Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately; namely, they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman‐Enskog expansion with Navier‐Stokes scaling. This extends the work of Golse and Saint‐Raymond [20,21] and Levermore and Masmoudi [28] to the case of a bounded domain. The case of a bounded domain was considered by Masmoudi and Saint‐Raymond [34] for the linear Stokes‐Fourier limit and Saint‐Raymond [41] for the Navier‐Stokes limit for hard potential kernels. Neither [34] nor [41] studied the damping of the acoustic waves. This paper extends the result of [34,41] to the nonlinear case and includes soft potential kernels. More importantly, for the Dirichlet boundary condition, this work strengthens the convergence so as to make the boundary layer visible. This answers an open problem proposed by Ukai [46]. © 2016 Wiley Periodicals, Inc.     

Initial‐Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

Initial‐boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so‐called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required to make the problem well‐posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so‐called global relation, and types of boundary conditions for which the global relation can be solved are called linearizable. For the defocusing nonlinear Schrödinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet‐to‐Neumann map supplied by the defocusing nonlinear Schrödinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial‐boundary value problem on the half‐line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter‐plane space‐time domain.     

随机环境中多类型接触过程的Hydrodynamic极限

本文首先构造环境中多类型接触过程，然后研究其Hydrodynamic行为，证明了该模型的Hydrodynamic极限是下列偏微分方程的解。