Local well-posedness for the fifth-order KdV equations on $\mathbb{T}$

Abstract

This paper is a continuation of the paper \emph{Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$}. In this paper, we consider the fifth-order equation in the Korteweg-de Vries (KdV) hierarchy as following: \begin{equation*} \begin{cases} \partial_t u - \partial_x^5 u + 30u^2\partial_x u + 20 u\partial_x u \partial_x^3u + 10u \partial_x^3 u = 0, \hspace{1em} (t,x) \in \mathbb{R} \times \mathbb{T}, u(0,x) = u_0(x) \in H^s(\mathbb{T}) \end{cases}. \end{equation*} We prove the local well-posedness of the fifth-order KdV equation for low regularity Sobolev initial data via the energy method. This paper follows almost same idea and argument as in the paper \emph{Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$}. Precisely, we use some conservation laws of the KdV Hamiltonians to observe the direction which the nonlinear solution evolves to. Besides, it is essential to use the short time $X^{s,b}$ spaces to control the nonlinear terms due to \emph{high $\times$ low $\Rightarrow$ high} interaction component in the non-resonant nonlinear term. We also use the localized version of the modified energy in order to obtain the energy estimate. As an immediate result from a conservation law in the scaling sub-critical problem, we have the global well-posedness result in the energy space $H^2$.