Classical Solutions of the Fornberg-Whitham Equation

TL;DR

This paper establishes the existence and uniqueness of classical solutions to the Fornberg-Whitham equation by analyzing its weak formulation within a Lagrangian framework, enhancing previous results in Sobolev and Besov spaces.

Contribution

It proves well-posedness of the Fornberg-Whitham equation in $C^1( eal)$ using a Lagrangian approach, providing a new classical solution framework.

Findings

Proves well-posedness in $C^1( eal)$ for the equation.

Constructs unique solutions dependent on initial data.

Improves upon previous results in Sobolev and Besov spaces.

Abstract

In this paper, we prove well-posedness in $C^1(\mathbb{R})$ (a.k.a. classical solutions) of the Fornberg-Whitham equation. To achieve this objective, we study its weak formulation under a Lagrangian framework. Applying the fundamental theorem of ordinary differential equations to the generated semi-linear system, we then construct a unique solution to the equation that is continuously dependent on the initial data. These results improve upon others in Sobolev and Besov spaces.