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.
Highlights
The intention of this review-type article is to put some of the key mathematical notions and solution results regarding the Fornberg–Whitham equation in a perspective with respect to each other and we will thereby strive to connect two so far largely parallel threads of research, because the same equation has been studied under the name of Burgers–Poisson equation
Neither do we attempt here to elaborate on the history and physics behind this model equation nor can we come anywhere near a complete overview of mathematical results from the more than 50 years of its analysis
We discuss here the Fornberg–Whitham equation as it was introduced by Whitham in [31, Eq (67)] as the integro-differential equation at the center of a shallow water wave model that is comparably simple and yet showed indications of wave breaking
Summary
The intention of this review-type article is to put some of the key mathematical notions and solution results regarding the Fornberg–Whitham equation in a perspective with respect to each other and we will thereby strive to connect two so far largely parallel threads of research, because the same equation has been studied under the name of Burgers–Poisson equation. Neither do we attempt here to elaborate on the history and physics behind this model equation nor can we come anywhere near a complete overview of mathematical results from the more than 50 years of its analysis. Our attention was restricted to results from work published at the time of writing and no systematic search of preprints was undertaken
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