Global well-posedness and blow-up of solutions for the Camassa–Holm equations with fractional dissipation

Abstract

Consideration in this paper is the effect of varying fractional dissipation with the dissipative operator power \( \gamma \ge 0\) on the well-posedness of the Camassa–Holm equations with fractional dissipation. It is shown that the zero-filter limit (\(\alpha \rightarrow 0\)) of the Camassa–Holm equation with fractional dissipation is the fractal Burgers equation. It is known that in the supercritical case \(\gamma \in [0, 1)\), the fractal Burgers equation blows up in finite time in \(H^s({\mathbb {R}})\) with \(s>\frac{3}{2}-\gamma \). It is established here that the dissipative Camassa–Holm equation is globally well-posed in the critical Sobolev space \(H^{\frac{3}{2}-\gamma }({\mathbb {R}})\) with the fractional parameter \(\gamma \in [\frac{1}{2}, 1)\). Moreover, it is also demonstrated that the solution of the dissipative Camassa–Holm equation blows up in finite time in the particular supercritical case when \( \gamma = 0\).