The Cauchy problem for higher-order modified Camassa-Holm equations on the circle

TL;DR

This paper establishes local well-posedness for higher-order modified Camassa-Holm equations on the circle with low regularity initial data, introducing new function spaces to overcome limitations of previous methods.

Contribution

It extends well-posedness results to lower regularity spaces by developing new function spaces and relationships, where standard Fourier methods fail.

Findings

Proves bilinear estimates are invalid for s<-+1 in Bourgain spaces.

Establishes local well-posedness for -j+3/2<s<-+1 with arbitrary initial data.

Introduces new function spaces to analyze the problem effectively.

Abstract

In this paper, we investigate the Cauchy problem for the shallow water type equation \begin{eqnarray*} u_{t}+\partial_{x}^{2j+1}u + \frac{1}{2}\partial_{x}(u^{2})+ \partial_{x}(1-\partial_{x}^{2})^{-1}\left[u^{2}+\frac{1}{2}u_{x}^{2}\right]=0 \end{eqnarray*} with low regularity data in the periodic settings. Himonas and Misiolek (Communications in Partial Differential Equations, 23(1998), 123-139.) have proved that the problem is locally well-posed for small initial data in H^{s}(\mathbf{T}) with s\geq-\frac{j}{2}+1,j\in N^{+} with the aid of the standard Fourier restriction norm method. To the best of our knowledge, there is no result of well-posedness about the problem when s<-\frac{j}{2}+1. In this paper, firstly, we prove that the bilinear estimate related to the nonlinear term of the equation in standard Bourgain space is invalid with s<-\frac{j}{2}+1. Then we prove that the Cauchy problem for the periodic shallow water-type equation is locally well-posed in H^{s}(\mathbf{T}) with -j+\frac{3}{2}< s<-\frac{j}{2}+1,j\geq2 for arbitrary initial data. The novelty is that we introduce some new function spaces and give a useful relationship among new spaces.