Abstract Cauchy-kovalevskaya Theorem In Gevrey Class And Its Application

In this paper,we study the well-posedness of nonlinear equation where the loss of derivatives occurs.We prove the existence and uniqueness of the equation’s solution in the Gevrey space with given initial values.We prove the abstract Cauchy-Kovalevskaya theorem forfor a class of nonlinear equations in the Gevrey space by energy method.This method is suitable for a large class of equations with derivative loss in fluid mechanics equations,such as Prandtl equation,MHD equation,especially boundary layer equation.The existence of equations is one of the basic problems in the field of partial differential equation.Whether the coefficients of the linear equations are constant or variable,there are perfect theories at present,which is attributed to early works and recent progresses of Hormander,Nirenberg,Treves,Lerner,Dencker and others.However,there is no general theories at present for the existence of nonlinear equations due to its complexity,except the classical Cauchy-Kovalevskaya theorem.However,the Cauchy-Kovalevskaya theorem is limited in analytic space,a natural question is whether it can be extended in a larger space.The equation studied in this paper is where Q∞,T=R×(0,T]and F(t,?xu)is analytic with respect to both x and t.The main tool is the combination of the classical energy method and abstract CauchyKovalevskaya theorem.The structure of this paper is as follows:In Chapter 1,we give an introduction to the background and development of the research and we list the main result of this paper.In Chapter 2,we present some preliminary knowledge.In Chapter 3,we prove the existence and uniqueness of the solution by using classical energy method and Cauchy-Kovalevskaya theorem in the Gevrey space.