The Cauchy problem for fractional Camassa-Holm equation in Besov space

TL;DR

This paper studies the fractional Camassa-Holm equation, proving local well-posedness in Besov spaces and demonstrating the analyticity of solutions with analytic initial data.

Contribution

It establishes local well-posedness in specific Besov spaces and proves analyticity of solutions for the fractional Camassa-Holm equation.

Findings

Well-posedness in Besov space $B^{s_0}_{2,1}$ for certain $s_0$ and $ u$

Analyticity of solutions with analytic initial data

Global in space, local in time analyticity results

Abstract

In this paper, we consider the fractional Camassa-Holm equation modelling the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. First, we establish the local well-posedness in Besov space $B^{s_0}_{2,1}$ with $s_0=2\nu-\frac 1 2$ for $\nu>\frac 3 2 $ and $s_0=\frac 5 2$ for $1<\nu\leq \frac 3 2 $. Then, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time.