Profile decompositions and Blowup phenomena of mass critical fractional   Schr\"odinger equations
  We study, under the radial symmetry assumption, the solutions to the fractional Schr\"odinger equations of critical nonlinearity in $\mathbb R^{1+d}, d \geq 2$, with L\'{e}vy index ${2d}/({2d-1}) < \al < 2$. We firstly prove the linear profile decomposition and then apply it to investigate the properties of the blowup solutions of the nonlinear equations with mass-critical Hartree type nonlineartity. 
Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon and Sanghyuk Lee
math.AP