Weak Convergence, 2-D Ideal Fluids and Harmonic Analysis

摘要吴主任研究员提出要研究两个问题：一个是广义的“齐化”问题，另一个是二维水波和欧拉方程。均匀化是研究材料微观结构的一种有效方法。解决这一问题的数学方法是弱收敛法，二维欧拉方程和二维重力波出现在海军研究和应用物理中。本文的目的是继续Coifman，Lions，Meyer和Semmes，Tartar和木拉提等人的工作，了解多线性算子和非线性算子的弱连续性、抵消性与Hardy空间之间的联系，找到非线性算子弱连续的充分条件；继续Nalimov，Yosihara和Walter Craig的工作，解决二维水波的存在唯一性问题。拟议工作的长期目标是发展更多关于非线性算子弱收敛的机制，并将结果和方法应用于解决非线性偏微分方程和应用科学中的公开问题；发展对二维和三维水波运动的完整理解，并发展一种可用于流体动力学数学理论其他领域的研究方法。所用的方法有调和分析、复分析、非线性偏微分方程组理论。

Abstract Wu The principle investigator proposes to study two problems: one is a generalized "homogenization" problem; another is the two dimensional water wave and Euler equations. Homogenization is an effective method used to study the micro-structure of materials. The mathematical method used to tackle the problem is the weak convergence method; 2-D Euler equations and 2-D gravity waves arise in the naval study and applied physics. The objective of the proposed works is to continue the line of research initiated by Coifman, Lions, Meyer and Semmes; and Tartar and Murat, and many others, to understand the link between weak continuity, cancelation property and Hardy spaces for multilinear operators and nonlinear operators, to find sufficient conditions for the weak continuity of nonlinear operators; and to continue the work of Nalimov, Yosihara and Walter Craig and solve the existence and uniqueness problem for 2-D water waves. The long term objective of the proposed works is to develop more machinery on the weak convergence of nonlinear operators and to apply the result and methods to solve open problems in nonlinear partial differential equations and applied sciences; and to develop a complete understanding of the motion of water waves, both two dimensional and three dimensional, and to develop a method which could be used in the study of other areas in the mathematical theory of fluid dynamics. The methods to be used are from harmonic analysis, complex analysis, the theory of nonlinear partial differential equations.