Refined wave breaking for the generalized Fornberg-Whitham equation

TL;DR

This paper analyzes wave breaking phenomena in a class of non-local dispersive equations related to the Burgers equation, constructing explicit blow-up solutions that exhibit shock-like singularities and self-similar behavior.

Contribution

It provides a detailed construction of blow-up solutions for the generalized Fornberg-Whitham equation, revealing their asymptotic self-similar structure and regularity properties.

Findings

Constructed explicit blow-up solutions with shock-like singularities.

Proved solutions converge to self-similar profiles of Burgers equation.

Established Hölder regularity of solutions at blowup points.

Abstract

This paper considers a class of non-local equations that are weakly dispersive perturbations of the inviscid Burgers equation, which includes the Fornberg-Whitham equation as a special case. We precise the known results on finite time blow-up (shock formation) by constructing a blowup solution which displays a `shock-like' singularity (called wave breaking) at one single point. Moreover, this solution converges asymptotically in the self-similar variables to a stable self-similar solution of the inviscid Burgers equation, and also possesses a H\"{o}lder $C^{1/3}$ regularity at the blowup point.