Sharp global well-posedness for a higher order Schrödinger equation

TL;DR

This work addresses the global well-posedness of a higher-order nonlinear Schrödinger equation on $\mathbb{R}$ in Sobolev spaces $H^s$ with $s>1/4$. The authors adapt the I-method, building almost conserved quantities by introducing Fourier multipliers and two modified energies, $E_1$ and $E_2$, constructed via a symbol $m(\xi)$ that is near identity at low frequencies and dampens high frequencies. They bound the time evolution of these energies, including control of the 4-linear and 6-linear contributions $\delta_4$ and $\delta_6$, via Littlewood–Paley decompositions, DMVT/TMVT estimates, improved Strichartz bounds, and a rescaling/iterative scheme to extend local results to all times. The main result is a sharp global well-posedness theorem for $s>1/4$, demonstrating global dynamics for the IVP and illustrating the effectiveness of the I-method in achieving global control at low regularity.

Abstract

Using the theory of almost conserved energies and the ``I-method'' developed by Colliander, Keel, Staffilani, Takaoka and Tao, we prove that the initial value problem for a higher order Schr\"odinger equation is globally well-posed in Sobolev spaces of order $s>1/4$.

Theorems & Definitions (25)

• Remark 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• proof
• Lemma 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof