On Well-posedness Of Some Nonlinear Evolution Equations In Water Wave

This paper studies two types of nonlinear evolution equations: generalized shallow water wave equation and Boussinesq system. The generalized shallow water equations studied in this paper is derived from common shallow water equations. They have great signi?cance in the waves, nonlinear optics, laser and plasma physics and extensive theoretical research value in mathematics. This article is to study the well-posedness of these equations, together with some results on ill-posedness and blow-up criteria.Boussinesq system models the atmospheric circulation, natural ventilation and central heating systems and other issues which are very useful, Is also one of the most important dynamics of geophysical model. In addition, two-dimensional Boussinesq system and three-dimensional Euler equations has a close association. Indeed, two-dimensional inviscid Boussinesq equations can be seen as a three-dimensional model of axisymmetric Euler equations when the vorticity away from the symmetry axis r = 0. To solve twodimensional Boussinesq equations will throw light on proving the well-posedness of Euler equations.In Chapter 2, we consider the Cauchy problem of the generalized CH equation with higher-order nonlinear terms?tu- utxx+(k + 2)ukux=(k + 1)uk-1uxuxx+ ukuxxx, t > 0, x ∈ R By using new Littlewood-Paley decomposition, Bony imitation skills, transport equation theory and Osgood lemma, we prove that when the initial value in the critical Besov spaces B3/22,1the Cauchy problem of GCH equation is well-posedness, whill it is ill-posed in B3/22,∞. Here Osgood Lemma give essential e?ect to make a breakthrough for the critical indicator case.In Chapter 3, we consider the Cauchy problem of the following two component integrable system????????mt+12(uv- uxvx)mx=-12((uv- uxvx)x-(uvx- uxv))m- bux,nt+12(uv- uxvx)nx=-12((uv- uxvx)x+(uvx- uxv))n- bvx,m = u- uxx, n = v- vxx By using the basic theoretical framework and supplementing a result on transport equation in low regularity initial space B-1/22,∞, we also get the existence and uniqueness of the solution for the Cauchy problem in the critical space B1/22,1. In addition, we prove thesolution is H¨older continuous dependencies with the initial value. In the fourth chapter,we continue to consider the generalized Camassa-Holm equation with multiple nonlinear terms and obtain similar results, which shows that our method has good applicability.In Chapter 5, we consider initial boundary value problem of the 2-dimensional viscous Boussinesq system:(Bν,0)??????tu- ν?u + u · ?u + ?π = θe2,(t, x) ∈ R+× ?,? · u = 0,?tθ + u · ?θ = 0,Here we try to consider the initial u0, θ0spaces as large as possible. We mainly use the energy method to obtain corresponding strong solution global existence and uniqueness.We also prove that the solution of the system Bν,0with initial and boundary value can be determined by the whole viscosity system Bν,κas the temperature viscosity tends to zero at the corresponding initial and boundary conditions in a way of strong convergence. We continue to study the Boussineq system with temperature viscosity, ?nd the corresponding space as large as possible and get a global existence result for the strong solution in Chapter 6. However, the previous methods in proving the uniqueness of the solution become invalid, thus it is necessary for us to ?nd new clues and this also shows that there are di?erences between the two viscous impact on the posedness for the Boussineq system.