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

## TL;DR

This paper proves global well-posedness for the defocusing mass-critical nonlinear fourth-order Schrödinger equation in certain Sobolev spaces below the energy space using the $I$-method and interaction Morawetz estimates.

## Contribution

It extends the understanding of the equation's well-posedness to lower regularity spaces below the energy space in dimensions 5 to 7.

## Key findings

- Global well-posedness established in $H^b3(\u211d^d)$ for specified $b3$ ranges.
- Uses $I$-method combined with interaction Morawetz estimate.
- Results cover dimensions 5, 6, and 7 with explicit regularity thresholds.

## Abstract

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 $H^\gamma(\mathbb{R}^d), 5\leq d\leq 7$ with $\gamma(d)<\gamma<2$, where $\gamma(5)=\frac{8}{5}, \gamma(6)=\frac{5}{3}$ and $\gamma(7)=\frac{13}{7}$.