KdV Limit For The One-dimensional Shallow Water Type System

In this dissertation,based on the shallow water equation,we investigate the KdV limit for the one-dimensional shallow water type equation with a special stress force by the singular perturbation theory,Gardner-Morikawa transform,and H?lder inequality,Cauchy inequality,Sobolev embedding theorem,Gronwall inequality.And we establish that the solutions of such shallow water type system converge to KdV equation as??0.This paper includes five parts:In part 1 we introduce the recent progress in the world of studies on water wave theory and limit theory,and what I have done,respectively.The second part is the introduction of some inequalities in Sobolev space and equations involved in the thesis.In part 3,the KdV equation is formally derived from one-dimensional shallow water type equation and we get the equations about(N,U)by the general form of perturbation expansion.In fourth part,by the structure characters of the equations about(N,U)and energy inequalities in Sobolev space,we obtain zero-order and first order of the energy inequalities about(N,U)and get the main theorem by using the Gronwall inequality.In fifth chapter,a summary of the author’s work in this paper along with an outlook for future research is given.