The Local Well-posedness Of The Two Dimensional Generalized Camassa-Holm-Kadomtsev-Petviashvili Equation

In the dissertation,it mainly researches the local well-posedness of the Cauchy problem of the two-dimensional generalized Camassa-Holm-KadomtsevPetviashvili(CH-KP)equation.In Chapter One,one introduces the research background of the water wave equation at first;Then one gives the research progress of the two-dimensional generalized CH-KP equation Cauchy problem and the main conclusions of this paper.Chapter Two is the preparatory knowledge part,which gives the relevant definitions and lemmas of this paper,including:product estimation,commutator estimation,Sobolev embedding,estimation of norm of composite functions and norm of intermediate variables and other basic knowledge.In Chapter Three,the existence and uniqueness of the solution to the Cauchy problem of the two-dimensional generalized CH-KP equation is studied mainly.There are three main steps.In the first step,one makes a priori estimation of the equivalent equation,then obtains the priori estimation of the solution under the higher-order norm combined with the Continuity method.In the second step,the existence of the solution is obtained by combining the approximation method and the compactness theory.The third step is to prove the uniqueness of solutions to generalized CH-KP Equation.Chapter Four studies the blow-up criterion and related theorems for the solution of the Cauchy problem of the two-dimensional generalized CH-KP equation.First,the Gagiardo-Nireberg inequality and the Young inequality of convolution are used to prove the blow-up criterion.Next,by constructing a function of time,the two blow-up theorems are proved by using the theory of particle trajectory equation and convolution estimation.Finally,combined with the previous research results and the research conclusions of this paper,future research directions are proposed.