On sharp global well-posedness and Ill-posedness for a fifth-order KdV-BBM type equation

TL;DR

This paper establishes sharp conditions for global well-posedness and ill-posedness of a fifth-order KdV-BBM type water wave equation, showing existence for data in Sobolev space $H^s$, $s\,\geq\,1$, and ill-posedness below that.

Contribution

It provides the first sharp thresholds for well-posedness and ill-posedness of this higher-order water wave model.

Findings

Global existence for data in Sobolev space $H^s$, $s\geq 1$.

Ill-posedness for $s<1$, flow-map discontinuity.

Sharpness of the well-posedness threshold.

Abstract

We consider the Cauchy problem associated to the recently derived higher order hamiltonian model for unidirectional water waves and prove global existence for given data in the Sobolev space $H^s$, $s\geq 1$. We also prove an ill-posedness result by showing that the flow-map is not continuous if the given data has Sobolev regularity $s< 1$. The results obtained in this work are sharp.