Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$

TL;DR

This paper establishes the local well-posedness of the fifth-order modified KdV equation on the torus for initial data in Sobolev spaces with regularity s > 2, using advanced Fourier analysis techniques.

Contribution

It provides the first low regularity well-posedness result for the fifth-order modified KdV equation on the periodic domain, employing the short-time Fourier restriction norm method.

Findings

Proves local well-posedness in H^s for s > 2

Utilizes the short-time Fourier restriction norm method

Employs frequency localized modified energy for estimates

Abstract

In this paper, we consider the fifth-order modified Korteweg-de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in $H^s(\mathbb T)$, $s > 2$, via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173 (2) (2008) 265--304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estimate. We remark that under the periodic setting, the integrable structure is very useful (but not necessary) to remove harmful terms in the nonlinearity and this work is the first low regularity well-posedness result for the fifth-order modified KdV equation.