From bipolar Euler-Poisson system to unipolar Euler-Poisson system in the perspective of mass

双极Euler-Poisson系统的零电子质量极限与准中性极限

Zero-electron-mass and quasi-neutral limits for bipolar Euler-Poisson systems

该研究针对双极Euler-Poisson方程，借助相对能量法结合椭圆理论、Riesz位势理论等工具，严格证明了零电子质量极限及零电子质量与准中性联合极限的合理性。

双极流体模型的松弛极限

The relaxation limit of bipolar fluid models

本研究针对双极流体模型，通过构建相对能量恒等式，证明了高摩擦区域下双极Euler-Poisson系统的耗散弱解收敛于双极漂移扩散系统的强解，建立了两类系统间的松弛极限关系。

具有空间相关阻尼的双极Euler-Poisson方程解的长时间行为

Large-time behavior of solutions to bipolar Euler-Poisson equations with space-dependent damping

本文通过加权能量方法证明，在空间相关阻尼下，一维双极Euler-Poisson方程Cauchy问题的小扰动初值存在唯一全局光滑解，并以与常数阻尼相同的代数速率收敛到稳态，且无需阻尼系数的小性条件。

Global well-posedness of the compressible bipolar Euler-Maxwell system in R^3

We first construct the global unique solution by assuming that the initial data is small in the H^3 norm but its higher order derivatives could be large. If further the initial data belongs to \Dot{H}^{-s} (0\le s<3/2) or \dot{B}_{2,\infty}^{-s} (0< s\le3/2), we obtain the various decay rates of the solution and its higher order derivatives. As an immediate byproduct, the L^p-L^2 (1\le p\le 2) type of the decay rates follow without requiring the smallness for L^p norm of initial data. In particular, the decay rate for the difference of densities could reach to (1+t)^{-13/4} in L^2 norm.

双极Euler-Poisson系统（其中一种流体无压力且无阻尼）的全局存在性与时间衰减

Global existence and time decay for a bipolar Euler-Poisson system with one pressureless and undamped fluid

本文研究三维双极Euler-Poisson系统，在一种流体无压力且无阻尼的非对称情形下，通过精细Green函数估计与高低频分解，证明了小扰动光滑解的全局存在性、唯一性及代数时间衰减。

Decay of the solution to the bipolar Euler-Poisson system with damping in $\mathbb{R}^3$

We construct the global solution to the Cauchy's problem of the bipolar Euler-Poisson equations with damping in $\mathbb{R}^3$ when $H^3$ norm of the initial data is small. If further, the $\dot{H}^{-s}$ norm ($0\leq s<3/2)$ or $\dot{B}_{2,\infty}^{-s}$ norm ($0<s\leq3/2$) of the initial data is bounded, we give the optimal decay rates of the solution. As a byproduct, the decay results of the $L^p-L^2$ ($1\leq p\leq2$) type hold without the smallness of the $L^p$ norm of the initial data. In particular, we deduce that $\|\nabla^k(ρ_1-ρ_2)\|_{L^2} \sim(1+t)^{-5/4-\frac{k}{2}}$ and $\|\nabla^k(ρ…

Optimal decay rate of the bipolar Euler-Poisson system with damping in $\mathbb{R}^3$

By rewriting a bipolar Euler-Poisson equations with damping into an Euler equation with damping coupled with an Euler-Poisson equation with damping, and using a new spectral analysis, we obtain the optimal decay results of the solutions in $L^2$-norm, which improve theose in \cite{Li3, Wu3}. More precisely, the velocities $u_1,u_2$ decay at the $L^2-$rate $(1+t)^{-{5}{4}}$, which is faster than the normal $L^2-$rate $(1+t)^{-{3}{4}}$ for the Heat equation and the Navier-Stokes equations. In addition, the disparity of two densities $ρ_1-ρ_2$ and the disparity of two velocities $u_1-u_2$ decay a…

Optimal large time behavior of the compressible Bipolar Navier--Stokes--Poisson system with unequal viscosities

This paper is concerned with the Cauchy problem of the 3D compressible bipolar Navier--Stokes--Poisson (BNSP) system with unequal viscosities, and our main purpose is three--fold: First, under the assumption that $H^l\cap L^1$($l\geq 3$)--norm of the initial data is small, we prove the optimal time decay rates of the solution as well as its all--order spatial derivatives from one--order to the highest--order, which are the same as those of the compressible Navier--Stokes equations and the heat equation. Second, for well--chosen initial data, we also show the lower bounds on the decay rates. Th…

Pointwise space-time estimates of 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities

Space-time behaviors for 3D compressible bipolar Navier-Stokes-Poisson system (BNSP) with unequal viscosities are given. The space-time estimate of electric field $\nablaϕ$ is the most important thing when deducing generalized Huygens' principle for BNSP since this estimate only can be obtained by $\nablaϕ=\frac{\nabla}Δ(ρ-n)$ from the Poisson equation. Thus, it requires to prove that the space-time estimate of $ρ-n$ only contains diffusion wave. The appearance of these unequal coefficients results that one cannot follow ideas for the special case, where the original system was rewritten as a …

Stability of nonlinear wave patterns to the bipolar Vlasov-Poisson-Boltzmann system

The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction wave for bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in [21, 23], we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as the applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave…

具有声速边界的半导体Euler-Poisson方程亚声速稳态解的最优正则性

Optimal regularity of subsonic steady-states solution of Euler-Poisson equations for semiconductors with sonic boundary

本文研究具有声速边界的半导体Euler-Poisson方程亚声速稳态解的最优正则性，利用比较原理和能量估计证明解属于$C^{\frac{1}{2}}[0,1]\cap W^{1,p}(0,1)$（$p<2$）且该结果最优，并分析了半导体松弛时间对声速点奇异性的影响。

Nonlinear structural stability and linear dynamic instability of transonic steady-states to a hydrodynamic model for semiconductors

For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, the existence of steady transonic shock solutions and C-smooth steady transonic solutions for Euler-Poisson Equations were established in [27] and [41], respectively. In this paper we further study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. When the C^1-smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the p…