The Properties Of The Solutions Of Two Types Of Benjamin-Ono Equation

There are a large number of nonlinear dispersion equations depending on continuous time in the field of fluid mechanics,which include the famous Benjamin-Ono equation.This equation is widely applied in the fields of biological marine engineering,meteorology and environment.This paper mainly focuses on the properties of the solution of the generalized Benjamin-Ono(g-BO)equation and the higher-order Benjamin-Ono(h-BO)equation.The first chapter mainly introduces the background and research status of two types of Benjamin-Ono equations,and presents the main research contents and common symbols in this paper.In chapter 2,we consider the continuity of the solution of the g-BO equation in soboiev space Hs(R)with S>3/2.Firstly,based on Sobolev basic inequality and the method of energy estimation,a prior estimate of the solution of the g-BO equation is obtained.By constructing the approximate solution,estimating the error between the approximate solution and the real solution and using the Sobolev interpolation theorem,it is proved that the solution of the g-BO equation is not uniformly continuous in Sobolev spaces Hs(R)with s>3/2.Simultaneously,the solution of the g-BO equation is Holder continuous in Hr-topology for all 0 ? r<s.In chapter 3,we mainly study the Gevrey regularity and analyticity of the data-to-solution map of the g-BO equation.At first,by means of the definition of Sobolev-Gevrey space,some basic properties of Sobolev-Gevrey space are derived.Then,applying the basic properties of Sobolev-Gevrey spaces and Ovsyannikov theorem,the data-to-solution map is Gevrey regularity and analyticity for the g-BO equation.In addition,a lower bound of the lifespan and the continuity of the data-to-solution map are also obtained.In chapter 4,we devote to investigate the persistence properties of the solution of the h-BO equation,i.e.,in weighted L2(R)spaces L?2=L2(R,?2dx),persistence properties of the solution of the h-BO equation is obtained by using a proper truncated weighting function and energy estimation method.Chapter 5 is a summary and expectation.