On the well-posedness of the Holm-Staley b-family of equations

TL;DR

This paper investigates the Holm-Staley b-family of equations within Sobolev spaces, demonstrating that their solution maps lack local uniform continuity regardless of the parameter value, using a geometric approach.

Contribution

It establishes the non-uniform continuity of the solution map for the Holm-Staley b-family in Sobolev spaces for all parameter values, a novel geometric analysis.

Findings

Solution map is nowhere locally uniformly continuous

Results hold for all parameter b

Uses geometric approach in Sobolev spaces

Abstract

In this paper we consider the Holm-Staley $b$-family of equations in the Sobolev spaces $H^s(\mathbb R)$ for $s > 3/2$. Using a geometric approach we show that, for any value of the parameter $b$, the corresponding solution map,$u(0) \mapsto u(T)$, is nowhere locally uniformly continuous.