Wave breaking for perturbed Burgers equations

Technical Summary: Wave Breaking for Perturbed Burgers Equations

Problem Statement
The paper investigates the phenomenon of wave breaking (shock formation) in a general class of perturbed Burgers-type equations. The primary equation of study is:
$$u_t + \left(\frac{u^2}{2}\right)_x = N[u],$$
where $N$ is a bounded linear operator mapping $H^1(\mathbb{R})$ to $L^2(\mathbb{R})$. While the classical Burgers equation is known to form steep gradients and break in finite time, the presence of a nonlocal source term $N$ can alter this behavior, potentially slowing down, accelerating, or preventing shock formation. The authors aim to establish simple, explicit, and verifiable criteria under which wave breaking occurs despite the presence of these nonlocal dispersive or dissipative effects.

The study covers a broad family of operators $N$ satisfying two key assumptions:

1. Commutation and Conservation: $N$ commutes with differentiation ($N[g'] = (N[g])'$) and satisfies an orthogonality condition ($\int N[g]g = 0$).
2. Norm Bounds: $N$ satisfies specific bounds involving $L^\infty$ and $L^2$ norms of the function and its derivatives, parameterized by constants $\eta, \alpha_i, \lambda_i$.

This framework encompasses several significant nonlocal models, including the Fractional KdV equation, the Whitham equation, and the Fornberg-Whitham equation.

Methodology
The core of the analysis relies on tracking the evolution of the slope of the solution along characteristic curves. The authors define $m(t) = -\inf_{x \in \mathbb{R}} u_x(t, x)$, which represents the maximum negative slope. Wave breaking is defined as the time $T^*$ when $m(t) \to +\infty$.

The proof strategy involves the following steps:

1. Characteristic Analysis: By differentiating the equation with respect to $x$ and evaluating along characteristics $\dot{\xi} = u$, the authors derive an ODE for the slope $v(t; \beta) = -u_x(t, \xi(t; \beta))$. This ODE takes the form $\dot{v} = v^2 - N[u_x]$.
2. Energy Estimates: The authors derive differential inequalities for the $L^2$ norms of the derivatives of $u$ (specifically $\|u_x\|_{L^2}$, $\|u_{xx}\|_{L^2}$, and $\|u_{xxx}\|_{L^2}$). These norms are shown to be controlled by $m(t)$.
3. Comparison Principle: A closed system of ODEs is constructed for $m(t)$ and auxiliary functions representing the $L^2$ norms of the derivatives. By utilizing the Gagliardo–Nirenberg interpolation inequality, the authors bound the nonlocal term $N[u_x]$ in terms of $m(t)$ and the $L^2$ norms.
4. Blow-up Criterion: The authors construct a comparison function $Z_\epsilon(t)$ based on the initial data and the $L^2$ norms. They prove that if the initial negative slope is sufficiently large relative to the $L^2$ norms of the higher derivatives and the operator constants, then $m(t)$ grows faster than $Z_\epsilon(t)$, leading to finite-time blow-up.

Key Results
The paper establishes Theorem 2.1, which provides explicit sufficient conditions for wave breaking. For a given parameter $\theta \in (0, \theta_0]$, if the initial data $\bar{u} \in H^3(\mathbb{R})$ satisfies a specific inequality relating the initial minimum slope $-\inf \bar{u}'$ to the $L^2$ norms of $\bar{u}''$ and $\bar{u}'''$, then wave breaking occurs at a time $T^*$ bounded by:
$$\frac{1}{(1+\theta)(-\inf \bar{u}') } \leq T^* \leq \frac{1}{(1-\theta)(-\inf \bar{u}') }.$$
This result demonstrates that nonlinear steepening dominates the nonlocal effects when the initial slope is sufficiently steep.

The paper further applies this main theorem to specific models:

• Fractional KdV Equation: For $\alpha \in (-1, -2/5)$, the authors derive a wave-breaking criterion (Proposition 3.1) that is simpler and more explicit than previous results in the literature.
• Whitham Equation: A specific criterion (Proposition 3.2) is derived, improving upon existing bounds found in prior works.
• Fornberg-Whitham Equation: The authors provide a comprehensive set of criteria (Proposition 3.3) covering various ranges of the parameter $s$ (including subcritical, critical, and supercritical cases). These results offer sharper bounds and cover cases where the kernel is non-integrable.

Additionally, Proposition 2.1 and Corollary 2.4 address cases where the operator $N$ is a convolution with an integrable kernel, showing that wave breaking occurs if the initial slope exceeds a threshold dependent on the $L^1$ norm of the kernel.

Significance and Claims
The authors claim that their proof is "both rigorous and straightforward," offering a unified framework for analyzing wave breaking across a diverse class of nonlocal equations. The primary contribution lies in the derivation of simple and explicit criteria that depend directly on the initial data's slope and derivative norms.

The paper asserts that these criteria yield precise bounds on the lifespan of classical solutions and identify the regimes where nonlinear steepening overcomes nonlocal dispersion. By providing these explicit conditions, the work simplifies the verification of wave breaking for models like the Fractional KdV, Whitham, and Fornberg-Whitham equations, offering improvements over previously established theorems in terms of simplicity and explicitness. The authors do not claim to resolve the global existence for all initial data but rather to delineate the specific conditions under which finite-time blow-up is guaranteed.