非线性偏微方程的拓扑几何学理论与方法

Based on stratification theory, we have following results: 1. The relationship between formal solvability of nonlinear partial differential equations (PDE) and project limit. The method of how to get a formal solution. Completing the research for initial and boundary value problems of Euler equations of inviscosity, incompressible fluid. 2. According to the evolution in 2000 and the requirements in practice, we roundly research following PDE’s initial problem: (i) A systematic study on two often used models of the system of Non-static Rotating Fluid is.being made. Several initial value problems and their well-posedness are discussed. The solution formulae is also given when the model is well-posed. (ii) Under the assumption of static equilibrium, we got the structure of solution space of general circulation equations in synoptic scale, and the criteria about the well-posedness of its initial value problem. We also compiled a symbolic computation program for its initial value problem. 3. Starting the research for well-posedness of baroclinic atmospheric equations’ initial problem. According to the well-posedness onditions, try to realize a general initial value problem which may conjectures future cases based on past cases. 4. Published a monograph titled “Introduction to Stratification Theory and Partial Differential Equations”(Chinese Edition) 5. Trained and training 7 post graduate students.

以分层理论为基础，研究非线性偏微分方程与常微分方程、非线性代数方程、函数方程之间内在的本质关系；方程的形式可解性与投影极限之间的关系；求形式解的方法与程序；不稳定方程的拓扑、几何性质以及数值不变量；鉴定流体力学、反应扩散中若干重要非线性偏微分方程的C^(k)稳定性与准确解等。