Almost Morawetz estimates and global well-posedness for the defocusing $L^2$-critical nonlinear Schr{\"o}dinger equation in higher dimensions

In this paper, we consider the global well-posedness of the defocusing, $L^{ 2}$ - critical nonlinear Schr{ \"o}dinger equation in dimensions $n \geq 3$. Using the I-method, we show the problem is globally well-posed in $n = 3$ when $s > { 2/5}$, and when $n \geq 4$, for $s > \frac{ n - 2}{ n}$. We combine energy increments for the I-method, interaction Morawetz estimates, and almost Morawetz estimates to prove the result.

Categories: math.AP