Refined wave breaking for the generalized Fornberg-Whitham equation

1. Problem Statement

The paper investigates the finite-time blow-up (wave breaking) of solutions to a class of non-local dispersive equations that serve as perturbations of the inviscid Burgers equation. The specific equation studied is:
$$u_t + u u_x - L u_x = 0$$
where $L$ is a Fourier multiplier operator defined by $\widehat{Lf}(\xi) = p(\xi)\hat{f}(\xi)$.

The authors focus on the Generalized Fornberg-Whitham equation, a specific case where the dispersion is given by:
$$K_\alpha = (I - \partial_x^2)^{(\alpha-1)/2} \implies p(\xi) = (1 + |\xi|^2)^{(\alpha-1)/2}$$
The primary regime of interest is $\alpha < 0$. In this regime, the dispersive effect is weak compared to the nonlinearity, leading to "hyperbolic-like" behavior where wave breaking (gradient blow-up while the solution remains bounded) is expected.

The Core Challenge: While previous works (e.g., [24], [27]) established the existence of wave breaking for $\alpha < 0$, they lacked precise descriptions of the blow-up mechanism. The goal of this paper is to provide a refined description of the blow-up, including:

• Precise blow-up time and location.
• The exact blow-up rate of the gradient.
• The regularity of the solution at the blow-up time (specifically Hölder continuity).
• Asymptotic convergence to a self-similar profile.

2. Methodology

The authors employ the modulated self-similar blowup technique, a sophisticated method recently developed for the Burgers equation and adapted here for non-local equations.

A. Self-Similar Transformation

The equation is transformed into self-similar variables to analyze the behavior near the singularity. The transformation is defined as:
$$s(t) = -\log(\tau(t) - t), \quad X(x, t) = \frac{x - \xi(t)}{(\tau(t) - t)^{3/2}}$$
$$u(x, t) = e^{-s/2} U(X, s) + \kappa(t)$$
Here, $\xi(t)$, $\tau(t)$, and $\kappa(t)$ are dynamic modulation variables representing the blow-up location, time, and wave amplitude, respectively. They are determined by imposing constraints on the solution $U$ at the origin ($X=0$):
$$U(0, s) = 0, \quad \partial_X U(0, s) = -1, \quad \partial_{XX} U(0, s) = 0$$

B. Decomposition of the Non-Local Operator

A critical technical difficulty arises because the Fourier multiplier $p(\xi)$ is inhomogeneous. The authors decompose the non-local operator $K_\alpha \partial_x$ into two parts based on the self-similar scaling:

1. Low-frequency part ($L$): Contains frequencies $|\xi| \lesssim e^{-3s/2}$. This part allows for strong temporal decay via Bernstein's inequality.
2. High-frequency part ($H$): Contains frequencies $|\xi| \gtrsim e^{-3s/2}$. This part behaves like a singular integral operator (related to the Hilbert transform) but possesses a specific integral kernel structure that allows for $L^\infty$ bounds without temporal decay.

C. Bootstrap Argument

The proof relies on a rigorous bootstrap argument. The authors assume a set of bounds (bootstrap assumptions) on the solution $U$ and its derivatives in different spatial regimes (near field, middle field, far field) and for different derivative orders. They then prove that these assumptions are self-consistent and can be closed, implying the existence of a solution satisfying these bounds up to the blow-up time.

Key steps in the analysis include:

• $L^2$ Estimates: Establishing bounds on low-order and high-order Sobolev norms to control the solution globally.
• $L^\infty$ Estimates: Using the transport structure of the equation along Lagrangian trajectories to derive pointwise bounds.
• Kernel Analysis: Detailed estimation of the integral kernels for the operator $H$ to handle the lack of temporal decay in the middle and far fields.

3. Key Contributions

1. Refined Blow-up Profile: Unlike previous results that only proved the existence of blow-up, this paper constructs a specific open set of initial data leading to a solution that converges asymptotically to a stable self-similar solution of the inviscid Burgers equation.
2. Handling Inhomogeneous Dispersion: The paper successfully addresses the technical challenges posed by the inhomogeneous symbol $p(\xi) = (1+|\xi|^2)^{(\alpha-1)/2}$, which differs from the homogeneous symbols used in fractional KdV studies. The decomposition into $L$ and $H$ operators is a novel adaptation for this specific equation.
3. Relaxed Hypotheses: Compared to similar works on the Burgers-Hilbert equation (e.g., [32]), the authors demonstrate that for $\alpha < 0$, fewer decay assumptions on the initial data are required to prove blow-up, as the dispersive effects are weaker.
4. Regularity Analysis: The paper precisely characterizes the regularity of the solution at the blow-up time, proving it is $C^{1/3}$ but not smoother.

4. Main Results (Theorem 2.1)

For $\alpha < 0$, there exists an open set of initial data in $H^5(\mathbb{R})$ such that the solution $u$ exhibits the following properties:

• Blow-up Time and Location: The solution blows up at a unique time $T^*$ and location $x^*$, with precise bounds:
  $$T^* \leq 2\varepsilon^{7/4}, \quad |x^*| \leq 3M\varepsilon$$
• Boundedness: The solution $u$ remains bounded in $L^\infty$ up to $T^*$.
• Gradient Blow-up Rate: The gradient $\partial_x u$ blows up at the rate:
  $$\frac{1}{3(T^* - t)} \leq \|\partial_x u(\cdot, t)\|_{L^\infty} \leq \frac{3}{T^* - t}$$
• Singularity Type: At the blow-up time $T^*$, the solution $u(\cdot, T^*)$ belongs to the Hölder space $C^{1/3}(\mathbb{R})$ and displays a cusp singularity at $x^*$.
• Asymptotic Convergence: In self-similar variables, the solution converges to a stable self-similar profile $U_\nu$ of the inviscid Burgers equation:
  $$\lim_{s \to \infty} \|U - U_\nu\|_{L^\infty} = 0$$
  where $U_\nu$ is a rescaled version of the ground state Burgers profile.

5. Significance

This work significantly advances the understanding of wave breaking in non-local dispersive equations.

• Theoretical Depth: It provides the first rigorous construction of a blow-up solution with a precise asymptotic profile for the Fornberg-Whitham equation in the $\alpha < 0$ regime.
• Methodological Innovation: The decomposition of the non-local operator into low and high-frequency components tailored to the self-similar scaling offers a robust framework for analyzing other equations with inhomogeneous dispersion.
• Physical Insight: The results confirm that when dispersion is weak ($\alpha < 0$), the dynamics are dominated by the hyperbolic nonlinearity, leading to shock formation that mimics the inviscid Burgers equation, despite the presence of non-local terms. The identification of the $C^{1/3}$ cusp singularity provides a precise mathematical description of the "breaking wave" shape.

In summary, the paper bridges the gap between the existence of blow-up and the detailed characterization of the singularity, establishing a stable, self-similar mechanism for wave breaking in the Generalized Fornberg-Whitham equation.