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In this paper, we consider the   $L^2$  -critical fractional Schrödinger equation   $iu_t-|D|^{\beta }u+|u|^{2\beta }u=0$   with initial data   $u_0\in H^{\beta /2}(\mathbb{R})$   and   $\beta $   close to   $2$  . We show that if the initial data have negative energy and slightly supercritical mass, then the solution blows up in finite time. We also give a specific description for the blow-up dynamics. This is an extension of the works of F. Merle and P. Raphaël for   $L^2$  -critical Schrödinger equations. However, the nonlocal structure of this equation and the lack of some symmetries make the analysis more complicated, hence some new strategies are required.

Blow-up Dynamics for L 2 -Critical Fractional Schrödinger Equations