Well-posedness for the fifth order KdV equation

TL;DR

This paper proves local well-posedness for the fifth order KdV equation with low regularity initial data by introducing a new function space and using the Fourier restriction norm method, overcoming derivative loss issues.

Contribution

It establishes well-posedness in a novel function space for low regularity data, extending previous results for the fifth order KdV equation.

Findings

Well-posedness in $H^{s,a}$ for specified $s$ and $a$ ranges.

Optimality of the regularity conditions in some cases.

Overcomes derivative loss with Fourier restriction norm method.

Abstract

In this paper, we establish the well-posedness for the Cauchy problem of the fifth order KdV equation with low regularity data. The nonlinear term has more derivatives than can be recovered by the smoothing effect, which implies that the iteration argument is not available when initial data is given in $H^s$ for any $s \in \mathbb{R}$. So we give initial data in $H^{s,a}=H^s \cap \dot{H}^a$ when $a \leq s$ and $a \leq 0$. Then we can use the Fourier restriction norm method to obtain the local well-posedness in $H^{s,a}$ when $s \geq \max\{-1/4, -2a-2 \}$, $-3/2<a \leq -1/4$ and $(s,a) \neq (-1/4,-7/8)$. This result is optimal in some sense.