Global existence for the defocusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space

Abstract

In this paper, we consider the defocusing mass-critical nonlinear fourth-order Schrödinger 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 $\gamma (7)=\frac{13}{7}$.

arXiv comment20 pages

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• Well-posedness of nolinear fractional Schrödinger and wave equations in Sobolev spaces
• Global well-posedness for a $L^2$-critical nonlinear higher-order Schrödinger equation