On the fifth order KdV equation: local well-posedness and lack of uniform continuity of the solution map

TL;DR

This paper establishes local well-posedness for the fifth order KdV equation in certain Sobolev spaces and demonstrates that its solution map is not uniformly continuous, highlighting nuanced stability properties.

Contribution

It proves local well-posedness for the fifth order KdV in $H^s$ for $s>5/2$ and shows the solution map lacks uniform continuity for $s>0$, advancing understanding of its mathematical behavior.

Findings

Well-posedness in $H^s$ for $s>5/2$

Solution map not uniformly continuous for $s>0$

Highlights stability limitations of the equation

Abstract

In this paper we prove that the fifth order equation arising from the KdV hierarchy $ \partial_tu + \partial_x^5u + c_1\partial_x u\partial_x^2u + c_2u\partial_x^3u = 0 $ is locally well-posed in $ H^s(\mathbb{R}) $ for $ s> 5/2. Also, we prove the solution map of the equation is not uniformly continuous for $s>0$.