Global well-posedness for the fractional NLS on the unit disk

In this talk, we discuss the cubic nonlinear Schr\"odinger equation with the fractional Laplacian on the unit disk. We show the global well-posedness for certain radial initial data below the energy space and establish a polynomial bound of the global solution. The result is proved by extending the I-method in the fractional nonlinear Schr\"odinger equation setting.