Solution concepts, well-posedness, and wave breaking for the Fornberg-Whitham equation

TL;DR

This paper reviews solution concepts and well-posedness for the Fornberg-Whitham equation, analyzing wave breaking, blow-up phenomena, and traveling wave solutions in the context of nonlinear PDEs.

Contribution

It provides a comprehensive comparison of strong and weak solutions, discusses blow-up behavior, and explores traveling wave solutions for the Fornberg-Whitham equation.

Findings

Wave breaking occurs for certain initial conditions.

Weak solutions include continuous traveling waves.

Semigroup methods relate to solution well-posedness.

Abstract

We discuss concepts and review results about the Cauchy problem for the Fornberg-Whitham equation, which has also been called Burgers-Poisson equation in the literature. Our focus is on a comparison of various strong and weak solution concepts as well as on blow-up of strong solutions in the form of wave breaking. Along the way we add aspects regarding semiboundedness at blow-up, from semigroups of nonlinear operators to the Cauchy problem, and about continuous traveling waves as weak solutions.