Concentration Of Blowup Solutions To The Dipolar Bose-Einstein Condensation

In quantum mechanics,the basic model describing Bose-Einstein condensation is Gross-Pitaevskii equation,while the model depicting dipolar Bose-Einstein condensation is Gross-Pitaevskii equation with a nonlocal nonlinearity.In this paper,starting from the rigorous mathematical theory of nonlinear partial differential equations,taking nonlinear interaction as the breakthrough point and using a series of modern variational methods,we construct new Gagliardo-Nirenberg inequalities.Moreover,we consider the sufficient conditions for the exis-tence of blow-up solutions and and concentration properties of blow-up solutions to the dipolar Bose-Einstein condensation.In chapter 1,we introduce the backgrounds and known results to the Bose-Einstein con-densation and the dipolar Bose-Einstein condensation.In chapter 2,we investigate the sufficient conditions for existence of blow-up solutions to the dipolar Bose-Einstein condensation.More narrowly,employing the profile decomposition theorem of bounded sequences in H¹ ? H⁽1/2 proposed by Zhu in[45],we firstly construct two refined variational inequalities.Secondly,we use the generalized Gagliardo-Nirenberg inequalities in H¹ to derive the sharp threshold for blow-up and global existence.Finally,we obtain a new sufficient condition for existence of blow-up solution to the dipolar Bose-Einstein condensation with electric potential by using the Virial identity and a special interpolation inequality.In chapter 3,a compactness lemma is utilized to prove that the blow-up solutions with bounded H^1/2 norm absolutely concentrate at least a fixed amount.