关于两类带电粒子传输模型非常值稳态解的扰动问题

This project will study the well-posedness on the perturbed problems of non-constant equilibrium solutions for two kinds of transmission models of the charged particles. The first one is called the Navier-Stokes-Poisson equations, which is often used to model the transport of charge carriers (electrons and holes) under the influence of self-consistent electrostatic field in semiconductors. The other model is a generalized Poisson-Nernst-Planck-Navier-Stokes equations, which can be used to simulate the transport of the crowded charged particles in a continuous medium. Such a model not only reveals the cross-diffusion phenomenon among the crowded particles, but also characterizes the macro-micro interactions of the medium and the charged particles. Under some specific conditions, these two models admit a non-constant equilibrium solution with zero velocity. Compared with the perturbation problems of the constant equilibrium solution, it is a relatively challenging work to study ones of the non-constant equilibrium solution of the above two models since there will be some new essential difficulties. In this project, some new methods and techniques are needed to be put forward, and even there will be a new theory on dealing with the perturbed problems of the non-constant equilibrium solution. So, this research will greatly enrich and develop the well-posedness theory of nonlinear partial differential equations in fluid dynamics.

本项目将研究两类带电粒子传输模型非常值稳态解的扰动问题的适定性。第一类模型称为Navier-Stokes-Poisson方程组，该方程组可以用来模拟半导体中载流子（电子和空穴）在自相容静电场影响下的运动传输。第二类模型称为广义Poisson-Nernst-Planck-Navier-Stokes方程组，该方程组可以用来模拟某种连续介质中稠密的带电粒子的运动传输，并且该广义方程组刻画了稠密粒子之间的交叉扩散现象和宏观介质与微观粒子之间的相互作用。在一些特定条件下，两类方程组都具有速度恒为零的非常值稳态解。对比常值稳态解的扰动问题，由于出现了一些新的本质性困难，研究这两类方程组非常值稳态解扰动问题的适定性，是一项富有挑战的工作。在研究过程中，需要提出一些新的方法和技术，甚至会形成研究非常值稳态解扰动问题的部分理论。因此，本项目的研究将会极大地丰富和发展流体力学中的非线性偏微分方程的适定性理论。

带电粒子的运动传输现象普遍存在于多种物理情境中，例如半导体、等离子体、化学电池和生物离子通道等等。本项目研究了多种带电粒子运动传输模型，包括带弹性的Navier-Stokes-Poisson方程组、非等温的Euler-Poisson方程组、双极非等熵的Euler-Maxwell方程组和一类广义的Poisson-Nernst-Planck-Navier-Stokes方程组，我们着重研究了这些模型稳态解的渐近稳定性性。特别地，对于它们的Cauchy问题，我们可以得到小扰动解及其高阶导数的最优代数衰减速率。这里的最优是指非线性问题的衰减率与线性问题的衰减率一致。