等离子体流体力学方程组解的适定性和渐近行为研究

等离子体在工业、能源、军事、航天等领域发挥着重要作用。Navier-Stokes-Poisson方程组和Navier-Stokes-Maxwell方程组是从流体力学角度刻画等离子体中带电粒子的运动规律的数学模型，其偏微分方程理论的研究是应用数学界的重要研究课题之一。方程组复杂的非线性结构增加了数学研究的难度，在解的适定性和渐近行为方面虽已有一些研究结果，但其理论研究并未完善，仍遗留了一些问题，如：在常值稳态解附近Navier-Stokes-Poisson方程组初边值问题经典解的指数衰减估计、Navier-Stokes-Maxwell方程组Cauchy问题经典解的最佳衰减估计和逐点估计，小能量弱解或经典解的适定性和渐近行为；非常值稳态解附近经典解的衰减估计等。本项目将采用谱分析和能量方法并探寻一些新方法来研究这些问题。这将会丰富和完善流体力学中非线性偏微分方程组的数学理论。

Plasmas play an important role in the field of industry, energy resource, military, aerospace. The Navier-Stokes-Poisson and Navier-Stokes-Maxwell equations are hydrodynamic models that can be used to describe the motion of the charged particles in plasmas. The study on these two partial differential equations is one of the important topics in applied mathematics. Because of the complex nonlinear structures, it's more difficult to study its mathematical theories. There have been some results about the well-posedness and the asymptotic behaviors of the solutions of these two equations. However, it still remains some problems to be solved, for example, the exponential decay estimates of the classical solution to the initial boundary value problem for the Navier-Stokes-Poisson equations and the optimal decay rates and pointwise estimates of the classical solution to the Cauchy problem for the Navier-Stokes-Maxwell equations with small initial data near constant equilibrium state; the well-posedness and the asymptotic behaviors of weak or classical solutions with the initial data that are of small energy; the well-posedness and decay estimates of the classical solutions near non-constant steady states. Therefore, in this project, we will explore the spectral analysis, energy methods and some new methods to study these problems. It will enrich and develop the mathematical theories of the nonlinear partial differential equations in fluid dynamics.