可压缩Euler方程组及其相关模型适定性和渐近极限

The fluid motion model involves a wide range of applications. The study of fluid models is of great significance both in theoretical exploration and in practical applications...This project focuses on the Euler equation, Euler-Poisson equation and the existence, uniqueness and asymptotic limit of the solution of the relevant model.This series of problems is the frontier and hot issue in the study of mathematical theory of nonlinear fluid evolution equations. This project will apply the variational and centralized methods to prove the existence of the equilibrium solution of these equations; Asymptotic limit of these models using linearization, construction methods, and approximation, boots and other techniques...Using the adjustment energy method, the spectral method, the Green function method, the Fourier split method combined with the high and low frequency decomposition to give the estimation and asymptotic limit of the solution to reveal the various states of the solution, thus the study of fluid models provides an entry point with physical and mathematical properties. We believe that it can not only enrich the theory of partial differential equations, but also provide theoretical support for numerical calculations and engineering applications.

流体运动模型涉及的范围非常广，对流体模型的研究无论是在理论探索还是实际应用中都有重要意义... 本项目主要针对Euler方程、Euler-Poisson方程及相关模型解的存在性、唯一性及渐近极限等问题进行全面而深入细致的研究，这一系列问题是当今非线性流体发展方程数学理论研究中的前沿和热点问题... 本项目将应用变分、集中紧方法证明这些方程平衡解的存在性；利用线性化、构造方法和逼近、靴带等技巧对这些模型的渐近极限进行研究；运用调整能量方法、谱方法、Green函数法、 Fourier分裂方法结合高低频分解给出解的极限情况和衰减估计，以揭示其解的各种性态，从而为流体模型的研究提供一个有物理直观和数学特性的切入点. 我们相信既能充实偏微分方程理论，也能为数值计算和工程应用提供理论支持.

流体模型涉及的范围非常广，对流体模型的研究无论是在理论探索还是实际应用中都有重要意义。项目主要围绕流体力学相关方程解的适定性及渐近极限等问题开展研究，如真空如何引起奇异性及高维时解的精细结构、对称约化、符号求解的研究等。具体来讲，我们的研究主要包括如下两个方面： . 1.研究了流体力学相关方程 (例如：Navier-Stokes方程组、Navier-Stokes-Maxwell方程组、微极流方程等) 解的存在性、稳定性和动力学行为；. 2. 研究了Euler方程组及相关模型的对称约化和符号求解。. 有关研究成果分别发表和接受发表在《J. Differential Equations》、《Journal de Mathématiques Pures et Appliquées》、《Comm. Math. Sci.》、《Nonlinear Anal. RWA》、《Indiana University Mathematics Journal》、《Journal of the London Mathematical Society》等期刊上。. 项目执行期间，召开了多次线上研讨会，并邀请多位同行专家在线上作学术交流。培养了5名博士研究生和4名硕士研究生。