Local well-posedness and global existence for the Popowicz system

TL;DR

This paper investigates the local well-posedness and global existence of solutions for the Popowicz system, an integrative model of Camassa-Holm and Degasperis-Procesi equations, in specific Besov spaces.

Contribution

It establishes new well-posedness results, a blow-up criterion, and global existence conditions for the Popowicz system in Besov spaces.

Findings

Local well-posedness in Besov spaces for s > max{2, 1/p + 3/2}

New blow-up criterion for solutions

Global existence results for various initial data

Abstract

Popowicz system, as the interacting system of Camassa-Holm and Degasperis-Procesi equations, has attracted some attention in recent years. In this paper, we first study the local well-posedness for the cauchy problem of Popowicz system in nonhomogeneous Besov spaces $B^s_{p,r}\times B^s_{p,r}$ with $s> \max\{2, \frac{1}{p}+\frac{3}{2}\}$ or $(s=2, 2\leq p \leq \infty, 1\leq r\leq 2)$. Moreover, a new blow-up criterion and global existence with different initial values are obtained.