Global Well-posedness for the fourth order nonlinear Schrödinger equations with small rough data in high demension

Abstract

For $n\geq 2$, we establish the smooth effects for the solutions of the linear fourth order Shrödinger equation in anisotropic Lebesgue spaces with $\Box_k$-decomposition. Using these estimates, we study the Cauchy problem for the fourth order nonlinear Schrödinger equations with three order derivatives and obtain the global well posedness for this problem with small data in modulation space $M^{9/2}_{2,1}({\Real^{n}})$.