Classical Solutions of the Fornberg-Whitham Equation

TL;DR

The paper proves local well-posedness of the Fornberg-Whitham equation in the classical space $C^1(\mathbb{R})$ by reformulating the PDE as a semi-linear ODE system in a Lagrangian framework and solving it via an ODE existence theorem. A diffeomorphism $\eta$ is constructed to recover the physical solution $u$ from the ODE variables, yielding a unique solution $u \in C([-T,T];C^1) \cap C^1([-T,T];C)$ with a lifespan $T = \frac{9}{100\|u_0\|_{C^1}}$ and a bound $\sup_{t\in[-T,T]} \|u(t)\|_{C^1} \le 2\|u_0\|_{C^1}$. The data-to-solution map is shown to be Hölder continuous from $C^\alpha$ to $C([-T,T];C^\alpha)$ for $0\le \alpha < 1$, and Lipschitz in the $C^0$ topology, extending classical well-posedness techniques to a nonlocal, nonintegrable dispersive model. These results provide a rigorous pathway to classical solutions and may inform similar analyses for other nonlinear dispersive equations.

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.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof