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

TL;DR

This paper establishes the optimal regularity threshold for well-posedness of fifth order KdV equations on periodic domains, using normal form reduction and cancellation properties to handle derivative losses.

Contribution

It proves unconditional well-posedness for fifth order KdV equations in $H^s$ with $s \,\geq\, 1$, introducing novel techniques to manage nonlinearities at this regularity.

Findings

Well-posedness holds for $s \ge 1$

Optimal regularity threshold established

Novel use of cancellation properties in analysis

Abstract

We consider the fifth order KdV type equations and prove the unconditional well-posedness in $H^s(\mathbb{T})$ for $s \ge 1$. It is optimal in the sense that the nonlinear terms can not be defined in the space-time distribution framework for $s<1$. The main idea is to employ the normal form reduction and a kinds of cancellation properties to deal with the derivative losses.