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

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 $s>35\mathrm{/}64$ in low modulation restricted norm spaces, as well as non-linear smoothing of order $\epsilon <\min (2(s-35\mathrm{/}64),1)$. 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 $s\ge 1$. We also show local well-posedness for the non-integrable fifth order KdV for $s>1\mathrm{/}2$, 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.