A Class Of Gross-Pitaevskii Equations With Rotation

We consider the Cauchy problem for the Gross-Pitaevskii(G-P)equation with an angular momentum rotation term under a partial confinement,which models Bose-Einstein condensation(BEC)with angular velocity rotation under an anisotropic magnetic trap and describes the mean field dynamics of rotating many-body bosons in a confinement trap.Beginning with the study of the BEC in dilute and thin atomic gases,more attention has been paid to dynamic phenomena related to the properties of superfluids.A notable superfluid feature is the quantification of vortices.Currently,in physical experiment,a laser beam with angular velocity rotation is applied to the BEC ground state to produce a harmonic anisotropic potential.Firstly,we discuss the sharp threshold for global existence in the mass-critical case of the G-P equation with rotation under a parial harmonic potential in two-dimensional space,and the sufficient conditions of blow up solutions.Then,the sharp threshold for global existence in the mass-critical case of the equation in N-dimensional(N_≥3)space,and the sufficient conditions of blow up solutions.Furthermore,we obtain the mass concentration properties of blow-up solutions.Specifically,firstly,by using the Hamilton conservations,the Virial identity,and the Sobolev embedding theorem,we get five sufficient conditions of blow up solutions,which further refines the three sufficient conditions for the nonlinear Schr?dinger equation given by Weinstein(1983).Secondly,inspiration of the discussion of the G-P equation with rotation under a harmonic potential,base on the variational characteristic of the ground state of the classic nonlinear scalar field equation,we establish the relationship between the classcial elliptic equation solution and our equation in corresponding space.So we get the Pohozaev identity of general elliptic equation.Then we construct new Gagliardo-Nirenberg inequalities,and we get the sharp threshold for global existence in the mass-critical case.Lastly,a refined compactness Lemma is utilized to prove that the profile decomposition of bounded sequence in corresponding space,then we discuss the mass concentration properties of blow-up solutions in N-dimensional(N_≥3)space.