The Cauchy Problem For A Modified Bl Equation And A Rigorous Link Between Modified Bl And Kp Equation

This paper consists of four chapters. The first chapter is the introduction. In thesecond chapter, we will study the existence and uniqueness of the local solution forthe Cauchy problem of the modified BL equation .In the third chapter, we will studythe existence and uniqueness of the the global solution to the Cauchy problem for themodified BL equation. In the fourth chapter,we will discuss the error estimate of theCauchy problem for the modified BL equation to the Cauchy problem for the KP- I andKP-Ⅱequation. The details are these:In the second chapter ,we discuss the existence and uniqueness of the local solutionfor the following Cauchy problem of the modified BL equation :whereε, a, 6, AandB are positive constants. u(x, y, t) denotes the unknown function.For this purpose, we will get the following equivalent form of the problem (1):where (?),(?).Then, using the contraction mapping principle and Sobolev interpolation estimation,we can prove the existence and uniqueness of the local solution for the Cauchyproblem (1), (2). The main result is the following:Theorem 1 Suppose that s≥2,φ∈H^s(R²),ψ∈H^s-1(R²).then the Cauchyproblem (1), (2) has a unique local solution u∈K^s([0,T⁰]), where [0, T⁰) is the maximal time interval of existence of u(x, y, t), Moreover, ifthenT⁰=∞.WhereK^s([0,T)=C([0,T],H^s(R²))∩C¹([0,T],H^s-1(R²))∩C²([0,T],H^s-2(R²))In Chapter 3, we prove the existence and uniqueness of the global solution to theCauchy problem (1), (2). The main results are the following:Theorem 2 Suppose that s≥3,φ∈H⁸(R²),ψ∈H^s-1(R²)and T₀＞0 is themaximal time interval of existence of the corresponding solution u(x, y, t) to the Cauchyproblem (1), (1). Then T₀=∞.By the error estimate ,In Chapter 4, our work is to link rigorously the Cauchyproblem for the modified BL equation to the Cauchy problem for the KP- I and KP-Ⅱequation.The main result of this work is the following:Theorem 3 Let there be givenp≥10 and T₀＞0 such that U is a solution ofthe KP equationsatisfyingThen there existε₀＞0 and a constant C₀＞0 such that for allε∈(0,ε₀],theequation (BL) has a unique solutionφ∈(?) of the formwhereρ(·,·,0)=ρ_t(·,·,0)=0. Furthermore,the following estimate holds:with C₀ independentε.In particular with C independent ofεWhere R^p([0,T)=C⁰([0,T],H^p(R²))∩C¹([0,T],H^p-3(R²))∩C²([0,T],H^p-6(R²)).K^s([0,T)=C([0,T],H^s(R²))∩C¹([0,T],H^s-1(R²))∩C²([0,T],H^s-2(R²)).