Fractional nonlinear Schr\"odinger and Hartree equations in modulation spaces

Abstract

We establish global well-posedness for the mass subcritical nonlinear fractional Schr\"odinger equation $$iu_t - (-\Delta)^\frac{\beta}{2} u+F(u)=0$$ having radial initial data in modulation spaces $M^{p,\frac{p}{p-1}}(\mathbb R^n)$ for $n \geq 2, p>2$ and $p$ sufficiently close to $2.$ The nonlinearity $F(u)$ is either of power-type $F(u)=\pm (|u|^{\alpha}u)\; (0<\alpha<2\beta / n)$ or Hartree-type $(|x|^{-\nu} \ast |u|^{2})u \; (0<\nu<\min\{\beta,n\}).$ Our order of dispersion $\beta$ lies in $(2n/ (2n-1), 2).$