Dynamics Of The Hartree Equations In The External Force Field

Hartree equation initially proposed by Hartree. As an important part of Hartree-Fock theory, it has a wide range of applications in diferent fields. Since1970s, it hasbeen widespread by mathematicians, fruitful results in theory have been achieved on theclassical Hartree equations.In this thesis, we try to extend the results on the free Hartree equations to the casein the extern fields. It is concerned with well-posedness, scattering, stability of standingwaves and coherent states etc. It is also the summarization and extraction of my papers[4] and [5] published during my ph. D. research.This thesis is divided into three parts: in the first part, we study the well-posednessand scattering for the Cauchy problem of the Hartree equations with potential. For details,1. we discuss the well-posedness of the Hartree equations with potential. The localexistence of the solution shall be established using Strichartz estimates. Then weshow the local Σ solution is global by means of blow-up alternative, mass conser-vation law, the identity the energy satisfied and Virial identity;2. applying the induction and bootstrap argument, we establish the exponential controlof the growth of higher order Sobolev norms for the global Σ solution;3. by establishing two identities, we prove the scattering of the defocusing Hartreeequation in energy subcritical case.In the second part, the standing waves of the Hartree equations with confining po-tential and the ground states of steady focusing equations are considered. We shall1. discuss the existence of the standing waves of defocusing Hartree equation;2. show the existence of ground states of the elliptic equations in the focusing case,meanwhile the regularity, decay, symmetry and positivity of the ground states shallbe discussed.3. establish the orbital stability of the standing waves by means of the stability of theabove ground states.In the last part, we shall construct the approximation solutions of the Hartree equations insemi-classical limit when the initial data is semi-classical wave packets (coherent states)and study the propagation of wave packets. For details,1. we get the approximation solutions in L2of the Hartree equations with smooth kernel. The result in [1] can be extended to general cases;2. the subcritical and critical cases for the Hartree quations with homogeneous k-ernel is studied. By the growth of the solutions and L2estimates, we show thepropagation of wave packets is the same as the linear case. Meanwhile, nonlinearsuperposition principle for two nonlinear wave packets is considered.