Global existence for the defocusing mass-critical nonlinear fourth-order Schr\"odinger equation below the energy space

In this paper, we consider the defocusing mass-critical nonlinear fourth-order Schr"odinger equation. Using the $I$-method combined with the interaction Morawetz estimate, we prove that the problem is globally well-posed in $H^{\gamma }({\mathbb{R}}^d),5\le d\le 7$ with $\gamma (d)<\gamma <2$, where $\gamma (5)=\frac{8}{5},\gamma (6)=\frac{5}{3}$ and $\gamma (7)=\frac{13}{7}$.