Dimension Reduction Of Nonlinear Hartree Equation With Strongly Confined Potential

The nonlinear Hartree equation with strongly confined potential is the equation describing the interaction between electrons in quantum mechanics.It is very important in quantum physics,plasma physics as well as chemistry.In this thesis,we consider dimension reduction of the nonlinear Hartree equation with strongly confined potential by two methods.In Chapter 1,the research background and status of the equation are introduced,and the main work of this thesis is given.In Chapter 2,the preparation of this thesis is given,and some definitions,lemmas,as well as notations are introduced.In Chapter 3,we consider dimension reduction of the nonlinear Hartree equation with anisotropic potential.Firstly,we obtain the low-dimensional modulation model of the nonlinear Hartree equation by using Hartree-Fock ansatz and projection method.And then,the error estimates for approximate solution of high-dimensional nonlinear Hartree equation is evaluated via Strichartz estimates,Hardy-Littlewood-Sobolev inequality as well as anisotropic Sobolev embedding.In Chapter 4,we consider dimension reduction of the nonlinear Hartree equation with strongly magnetic confined potential.According to averaging theory,we obtain a lowdimensional averaged approximate model of nonlinear Hartree equation.Furthermore,we study local well-posedness of the averaged approximate model by adapting the fixed point argument.Thus,the error estimates for the approximate solution of the nonlinear Hartree equation can be constructed by the local Lipschitz continuity of the Hartree term and Gronwall inequality.In Chapter 5,we show summary and prospect of this thesis.