动力学方程及相关模型的稳定性研究

The classical Vlasov-Poisson-Boltzmann system and related models describe dynamics of the weakly ionized collisional plasma interacting with its own electrostatic field as well as its grazing collisions.The term "fourth state of matter" is often used to describe the plasma state. Much of the understanding of plasmas has come from the pursuit of controlled nuclear fusion and fusion power, for which plasma physics provides the scientific basis. The main purpose of this project is to study the stability of one specie (two species) Vlasov Poisson Boltzmann system , Vlasov Poisson Landau system and Navier Stokes Poisson system (bipolar). More precisely, we will focus on the existence and the stabiltiy of the nontrivial solutions of the above system. Assume the electric potential takes two distinct constant states at infinity, we will seek the nontrivial solutions which are the perturbations of the profile of the corresponding Riemann problem of the Euler equations or Navier-Stokes system. In summary, we will ivestigate the global stability of the rarefaction wave and travelling wave of the Vlasov Poisson Boltzmann、Vlasov Poisson Landau and Navier Stokes Poisson systems. We hope that our study in this project will be helpful to illustrate the wave phenomenon of the plasma.

Vlasov-Poisson-Boltzmann 方程组及相关方程近似的描述了弱电离化的等离子体（如离子、电子等）在电场作用下的动力学行为。等离子体作为宇宙物质的"第四态"，对其认识和理解主要来源于对核聚变以及核电站的可控性研究。本项目将主要研究Vlasov-Poisson-Boltzmann （单族、双族）、Vlasov-Poisson-Landau、Navier-Stokes-Poisson（单极、双极）等方程组的解的稳定性问题。更准确的说我们将关注上述方程组非平凡解的稳定性，即给定电场在正负无穷远处有两个不同的边值，我们希望上述方程组的非平凡解是其对应Euler方程或Navier-Stokes方程黎曼问题解的一个扰动，基于此设想我们将研究上述方程组的稀疏波、粘性激波等非平凡解的整体稳定性。这些研究将有助于等离子体研究中波现象的物理解释。

本项目按照计划书中的研究课题稳定有序的推进，在国家自然科学基金的强力支持下，取得了预期满意的成果。Vlasov-Poisson-Boltzmann 方程组及相关方程近似的描述了弱电离化的等离子体（如离子、电子等）在电场作用下的动力学行为。等离子体作为宇宙物质的“第四态”，对其认识和理解主要来源于对核聚变以及核电站的可控性研究。本项目我们主要研究了Vlasov-Poisson-Boltzmann （单族、双族）、Vlasov-Poisson-Landau、Navier-Stokes-Poisson（单极、双极）等方程组的解的稳定性问题、Boltzmann方程在临界正则性空间的适定性问题以及Boltzmann方程的初边值问题。更准确的说我们重点研究了上述动理学方程非平凡解的稳定性，即给定电场在正负无穷远处有两个不同的边值，通过构造其对应Euler方程或Navier-Stokes方程黎曼问题的背景解，我们严格证明了上述方程组的稀疏波、接触间断以及粘性激波等非平凡解的整体稳定性，这些研究将有助于等离子体研究中波现象的物理解释。此外我们还研究了Boltzmann方程在临界正则性空间的适定性问题，这一问题也与Boltzmann方程的初边值问题有关，因为通常来讲，Boltzmann方程的初边值问题的解具有有限的正则性，这两类问题的研究将有助于我们进一步研究Vlasov-Poisson-Boltzmann 方程组及相关方程在一般有界区域上的初边值问题。上述相关成果已经正式发表在Archive for Rational and Mechanics and Analysis、Journal of Functional Analysis、SIAM Journal on Mathematical Analysis、Journal of Differential Equations、Math. Models Methods Appl. Sci.、Kinetic and Related Models 等国际主流数学杂志。