Global Well-posedness for the fourth order nonlinear Schr\"{o}dinger equations with small rough data in high demension

TL;DR

This paper proves the global well-posedness of high-dimensional fourth order nonlinear Schrödinger equations with small, rough initial data by establishing smooth effects in anisotropic Lebesgue spaces and using modulation space estimates.

Contribution

It introduces new smooth effect estimates for the linear equation and applies them to demonstrate global well-posedness for the nonlinear problem with small initial data.

Findings

Global well-posedness established for small data in modulation space.

Smooth effects for solutions in anisotropic Lebesgue spaces proved.

Applicable to high-dimensional fourth order nonlinear Schrödinger equations.

Abstract

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