Blow-up dynamics for $L^2$-critical fractional Schrödinger equations

Abstract

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

41 pages. Part of the proof has been reorganized