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.
