Global well-posedness for the mass-critical nonlinear Schrödinger equation on

We consider the global well-posedness for the Cauchy problem of the mass-critical nonlinear Schrödinger equations in the periodic case. We show that it is globally well-posed in for any s>2/5. This improves the related work of Bourgain (2004) [2]. The key point is that we combine I-method with the resonant decomposition, which is developed in Colliander et al. (2008) [9], Li et al. (2011) [15], Miao et al. (2010) [16]. Another new ingredient here is that we obtain a bilinear Strichartz estimates in the periodic case which improves slightly the result given in De Silva et al. (2007) [11].