Global Behavior Of Finite Solutions To The Dipolar Quantum Gas

The dipole Bose-Einstein condensate is a condensed state composed of parti-cles with electric or magnetic dipole moment.And it is a macroscopic wave made out of a large number of particles with the same quantum state.In this thesis,we study the nonlinear Schr¨odinger equation describing the dipolar quantum gases,and show its global behavior of finite energy solutions including their scattering theory and weak blowup.For the scattering theory,inspired by the Dodson-Murphy,we adopt a simpler approach to avoid the concentration compactness principle.However,the dipolar interaction makes it impossible to estimate the interaction Morawetz inequality directly.And the dipole kernel is bounded only in the L~2-norm which increases difficulties in estimating the space-time norms.In this thesis,applying traditional Morawetz estimates,interaction Morawetz estimates and bootstrap argument,we show a new proof of scattering.According to dichotomy,in the energy space,the solution either blows up in finite time,or exists globally and diverges to infinite when time goes to infinite.Using the Gagliardo-Nirenberg inequality,we obtain the sharp threshold of the global existence and blowup.The concentration compactness principle is applied to construct a critical solution.By analyzing the localization property of the critical solution,we prove the weak blowup in the end.