Global Well-posedness for the Biharmonic Quintic Nonlinear Schr\"odinger Equation on $\mathbb{R}^2$

TL;DR

This paper establishes global well-posedness for the 2D quintic defocusing biharmonic Schr"odinger equation in Sobolev spaces with regularity between 8/7 and 2, using the $I$-method to control energy growth.

Contribution

It extends the global well-posedness results to lower regularity Sobolev spaces for the 2D biharmonic Schr"odinger equation using the $I$-method.

Findings

Proves global well-posedness in $H^s(\

for 8/7 < s < 2.

Introduces a modified energy functional that is almost conserved over time.

Abstract

We prove that the Cauchy problem for the 2D quintic defocusing biharmonic Schr\"odinger equation is globally well-posed in the Sobolev spaces $H^s(\mathbb{R}^2)$ for $\frac{8}{7}<s<2$. Our main ingredient to establish the result is the $I$-method of Colliander-Keel-Staffilani-Takaoka-Tao \cite{colliander2002almost} which is used to construct the modified energy functional that is almost conserved in time.