Well-posedness and Continuity Properties of the Fornberg-Whitham Equation in Besov Spaces

TL;DR

This paper establishes well-posedness, continuity properties, and analytic solutions for the Fornberg-Whitham equation in Besov spaces, including sharpness results and blow-up criteria, advancing the mathematical understanding of this nonlinear PDE.

Contribution

It proves well-posedness in Besov spaces, demonstrates the non-uniform continuity of the data-to-solution map, and establishes a Cauchy-Kowalevski theorem for real analytic solutions.

Findings

Well-posedness in Besov spaces $B_{2,r}^s$

Non-uniform continuity of the data-to-solution map

Existence of real analytic solutions and blow-up criteria

Abstract

In this paper, we prove well-posedness of the Fornberg-Whitham equation in Besov spaces $B_{2,r}^s$ in both the periodic and non-periodic cases. This will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data-to-solution map provided that the initial data belongs to $B_{2,r}^s $. We also establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from any bounded subset of $B_{2,r}^s$ to $C([-T,T]; B^s_{2,r})$. Furthermore, we prove a Cauchy-Kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow-up criterion for solutions.