On the Cauchy Problem for the Fractional Camassa-Holm Equation

TL;DR

This paper proves local well-posedness for the fractional Camassa-Holm equation's Cauchy problem using Kato's semigroup approach, applicable to initial data in certain Sobolev spaces, advancing understanding of wave propagation in elastic media.

Contribution

It establishes local well-posedness results for the fractional Camassa-Holm equation with initial data in Sobolev spaces, employing Kato's semigroup method.

Findings

Cauchy problem is locally well-posed for s > 5/2

Uses Kato's semigroup approach for quasilinear equations

Models wave propagation in nonlocal elastic media

Abstract

In this paper, we consider the Cauchy problem for the fractional Camassa-Holm equation which models the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. Using Kato's semigroup approach for quasilinear evolution equations, we prove that the Cauchy problem is locally well-posed for data in $H^{s}({\Bbb R})$, $s>{\frac{5}{2}}$.