Integral Equations With Singularities And Falkner-Skan Problems

Falkner-Skan problems are used to describe the two-dimensional flow of a slightly viscous fluid past wedge shaped bodies or a flat plate in boundary layer theory of fluid dynamics, which are of great importance in the study of many fields such as incompressible and compressible boundary layer theory.This thesis establishes integral equations or systems with singularities that are equivalent to the Falkner-Skan problems and fixed theorems in Banach space, Helly selection principle of bounded variation functions and some special analytic techniques are used to treat these integral equations or systems. By this new approach of the equivalence, many new results such as upper and lower bounds of the critical value, the shear stress and the existence and nonexistence of the Falkner-Skan problems are obtained. It seems to be very difficult that the theory and methods of "shooting methods", upper and lower solutions, topological transversality theorem and cone mappings are applied to obtain them. These results will be helpful to study boundary layer theory.Main results of this thesis are as follows:(1) Integral equations or systems singularities are established, which are equivalent to the Falkner-Skan problems.(2) Upper and lower bounds of the critical value and the shear stress involved in the Falkner-Skan problems are estimated analytically.(3) Existence and nonexistence of solutions of the Falkner-Skan problems are studied.