带物理小参数的Euler-Poisson方程组及Euler-Maxwell方程组的整体收敛性和稳定性

本项目中我们考虑两类带多个物理小参数的等离子体的数学模型，它们分别是高维空间上的Euler-Poisson方程组及Euler-Maxwell方程组。国内外学者对这两个方程组已有很多研究成果，处理的问题主要集中在常数稳定态附近光滑解关于时间的整体存在性及长时间行为，及方程组在参数趋于零时关于时间的局部收敛性。我们主要研究近十年来国内外学者正在考虑的一些新问题，它涉及到方程组在参数趋于零时关于时间的整体收敛性，光滑解关于小参数的一致整体存在性和在非常数稳定态附近的整体稳定性等。这两类模型可以归入高维部分耗散的一阶拟线性对称双曲型方程组的框架，我们的研究方法包含对称化方法，能量估计及耗散估计方法，带权的Sobolev空间，紧致性理论和测度理论等。此外，我们也研究一类强碰撞模型，它们也是一阶拟线性对称双曲型方程组，但是并不部分耗散。 本项目旨在得到一些有意义的原创性的研究结果。

In this project，we consider two kinds of mathematical models arising in plasma physics with small parameters. They are Euler-Poisson systems and Euler-Maxwell systems in several space dimensions. For these systems there exist a lot of research results. The most of them concentrate on the problems of global existence of smooth solutions around constant equilibrium states with long time behavior, and local-in-time convergence of the systems as parameters go to zero. We mainly study problems raised recently for these two kinds of models. They concern the global-in-time convergence of the systems as parameters go to zero, the uniform global existence of solutions with respect to the parameters and the stability of solutions around non-constant equilibrium states. These systems belong to the class of first-order quasi-linear symmetrizable hyperbolic systems which are partially dissipative. Our research methods include the choice of symmetrizer, energy estimates and dissipation estimates, weighted Sobolev spaces, compactness theory and measure theory. Moreover, we study a kind of strong collision models which are also first-order quasi-linear symmetrizable hyperbolic systems but are not partially dissipative. The goal of our project is to obtain significant original research results.