The Blowup Conditions Of The Nonlinear Schr?dinger Equations With Magnetic Field

The nonlinear Schr?dinger equation with magnetic field is an equation in quantum mechanics which describs the state of particle motion in non-relativistic situations.And it is one of the important objects of partial differential equations.In this thesis,we mainly study the blowup criteria for the solution of the nonlinear Schr?dinger equation with magnetic field in the three-dimensional space.In the case that the equation with harmonic potential,we analyze the variational structure of the equation,and then establish the corresponding invariant evolution flows.Thus the sharp threshold of global existence and blowup for the equation are obtained in two conditions respectively.In the case of cubic nonlinear term,by analyzing some invariants of the equation and combining the ground-state solutions of cubic nonlinear elliptic equations,the sufficient conditions of blowup for the solutions of the equations are given.Firstly,we introduce the known results of this equation and the main results in this thesis.Secondly,we give prelimaries.Finally,the threshold of the solution blowup and the global existence are studied,and the blowup solution of the equation with cubic nonlinear term in three-dimensional space is obtained.