Well-posedness for the Non-integrable Periodic Fifth Order KdV in Bourgain Spaces

TL;DR

This paper establishes well-posedness results for a non-integrable fifth order KdV equation using advanced harmonic analysis techniques, extending known results and providing new bounds on solution complexity.

Contribution

It introduces differentiation-by-parts in Bourgain spaces to prove well-posedness for non-integrable fifth order KdV, including unconditional and global results, and explores nonlinear smoothing effects.

Findings

Well-posedness for s > 35/64 in Bourgain spaces.

Unconditional well-posedness for s > 1.

Global well-posedness for the integrable case at s ≥ 1.

Abstract

We study well-posedness for a non-integrable generalization of the fifth order KdV, the second member in the KdV heirarchy. In particular, we use differentiation-by-parts to establish well-posedness for $s> 35/64$ in low modulation restricted norm spaces, as well as non-linear smoothing of order $\varepsilon < \min(2(s-35/64), 1)$. As corollaries, we obtain unconditional well-posedness for the non-integrable fifth order KdV for $s > 1$ and global well-posedness for the integrable fifth order KdV for $s\geq 1$. We also show local well-posedness for the non-integrable fifth order KdV for $s > 1/2$, contingent upon the conjectured $L^8$ Strichartz estimate. As an application of the nonlinear smoothing we obtain non-trivial upper bounds on the upper Minkowski dimension of the solution to the non-integrable fifth order KdV.