Global Well-posedness for The Fourth-order Nonlinear Schrödinger Equations on ${\mathbb{R}}^2$

We study the global well-posedness of the two-dimensional defocusing fourth-order Schrödinger initial value problem with power type nonlinearities $\mathrm{\mid }u{\mathrm{\mid }}^{2k}u$ where $k$ is a positive integer. By using the $I$-method, we prove that global well-posedness is satisfied in the Sobolev spaces $H^s({\mathbb{R}}^2)$ for $2-\frac{3}{4k}<s<2$.