Global well-posedness for the fifth-order KdV equation in $H^{-1}(\mathbb{R})$

Abstract

We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in $H^{-1}(\mathbb{R})$. By comparison, the optimal regularity for well-posedness on the torus is known to be $L^2(\mathbb{R}/\mathbb{Z})$. In order to prove this result, we develop a strategy for integrating the local smoothing effect into the method of commuting flows introduced previously in the context of KdV. It is this synthesis that allows us to go beyond the known threshold on the torus.

References in corpus (3)

• Conserved energies for the cubic NLS in 1-d
• Invariance of white noise for KdV on the line
• On the extension of the frequency maps of the KdV and the KdV2 equations

Cited by in corpus (5)

• On the well-posedness problem for the derivative nonlinear Schrödinger equation
• Sharp well-posedness results of the Benjamin-Ono equation in $H^s(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution
• Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$
• KdV on an incoming tide
• Long-time Asymptotic Behavior of the Fifth-order Modified KdV Equation in Low Regularity Spaces