Cancellation properties and unconditional well-posedness for fifth order modified KdV type equations with periodic boundary conditions

TL;DR

This paper establishes the unconditional local and global well-posedness of fifth order modified KdV equations with periodic boundary conditions in certain Sobolev spaces, using normal form reduction and cancellation properties.

Contribution

It proves unconditional well-posedness for non-integrable fifth order mKdV equations in $H^s(\mathbb{T})$ for $s \geq 3/2$, and global results at $s=2$.

Findings

Unconditional well-posedness for $s \geq 3/2$

Global well-posedness at $s=2$

Use of normal form reduction and cancellation properties

Abstract

We prove the unconditional well-posedness result for fifth order modified KdV type equations in $H^s(\mathbb{T})$ when $s \geq 3/2$, which includes non-integrable cases. By the conservation laws, we also obtain the global well-posedness result when $s = 2$, which also includes non-integrable cases. The main idea is to employ the normal form reduction and a kind of cancellation properties to deal with the derivative losses.