Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation
Sung-Jin Oh, Federico Pasqualotto
TL;DR
The paper proves the existence of gradient blow-up with bounded amplitude for a broad class of dispersive and dissipative perturbations of Burgers, including the fractional KdV, fractal Burgers, and Whitham equations, by leveraging a modulation-theory framework around Burgers self-similar profiles. A robust weighted $L^{2}$ approach is developed to control the spatial behavior in self-similar variables and to obtain sharp Hölder regularity up to blow-up; the analysis also extends to excited self-similar profiles, revealing a stability-perturbation correspondence: profiles with slower concentration are more tolerant to equation perturbations. The main results establish finite-time blow-up for all admissible $\alpha,\beta$ with $\alpha,\beta<\frac{2k}{2k+1}$, provide a precise asymptotic description near the singularity, and, in the case $k=1$, yield stability under initial-data perturbations for $\alpha<\frac{2}{3}$ and $\beta<\frac{2}{3}$. The findings offer a detailed construction of shock-like singularities in nonlocal models of water waves and illuminate how excited self-similar profiles can govern blow-up dynamics under dispersive/dissipative perturbations, with potential implications for broader singularity formation problems in fluids.
Abstract
We consider a class of dispersive and dissipative perturbations of the inviscid Burgers equation, which includes the fractional KdV equation of order $α$, and the fractal Burgers equation of order $β$, where $α, β\in [0,1)$, and the Whitham equation. For all $α, β\in [0,1)$, we construct solutions whose gradient blows up at a point, and whose amplitude stays bounded, which therefore display a "shock-like" singularity. We moreover provide an asymptotic description of the blow-up. To our knowledge, this constitutes the first proof of gradient blow-up for the fKdV equation in the range $α\in [2/3, 1)$, as well as the first description of explicit blow-up dynamics for the fractal Burgers equation in the range $β\in [2/3, 1)$. Our construction is based on modulation theory, where the well-known smooth self-similar solutions to the inviscid Burgers equation are used as profiles. A somewhat amusing point is that the profiles that are less stable under initial data perturbations (in that the number of unstable directions is larger) are more stable under perturbations of the equation (in that higher order dispersive and/or dissipative terms are allowed) due to their slower rates of concentration. Another innovation of this article, which may be of independent interest, is the development of a streamlined weighted $L^{2}$-based approach (in lieu of the characteristic method) for establishing the sharp spatial behavior of the solution in self-similar variables, which leads to the sharp Hölder regularity of the solution up to the blow-up time.