Blow-up Dynamics for L2-Critical Fractional Schrödinger Equations

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.