Global Well-posedness for The Fourth-order Nonlinear Schr\"{o}dinger Equations on $\mathbb{R}^{2}$

TL;DR

This paper proves the global well-posedness of a two-dimensional defocusing fourth-order nonlinear Schrödinger equation with power nonlinearities in certain Sobolev spaces using the $I$-method.

Contribution

It establishes global well-posedness results for the 2D defocusing fourth-order NLS with power nonlinearities in Sobolev spaces, extending previous understanding.

Findings

Global well-posedness in $H^{s}(R^{2})$ for $2-rac{3}{4k}<s<2$

Application of the $I$-method to higher-order Schrödinger equations

Extension of well-posedness results to a range of Sobolev spaces

Abstract

We study the global well-posedness of the two-dimensional defocusing fourth-order Schr\"odinger initial value problem with power type nonlinearities $\vert u\vert^{2k}u$ 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 $2-\frac{3}{4k}<s<2$.