Dynamical behaviors of blowup solutions in trapped quantum gases: Concentration phenomenon

The current paper contemplates the blowup dynamics in trapped dipolar quantum gases. More precisely, employing the profile decomposition of bounded sequences in H˙1∩H˙12, we firstly construct related variational problems and derive two refined Gagliardo-Nirenberg inequalities. Secondly, a compactness lemma is utilized to prove that the blowup solutions with bounded H˙12 norm and bounded L3 norm absolutely concentrate at least a fixed amount.