Asymptotically self-similar shock formation for 1d fractal Burgers equation

TL;DR

The paper proves that for the fractal Burgers equation with $0<\alpha<\tfrac{1}{3}$, there exist open sets of $H^6$ initial data that generate a finite-time gradient shock, with the singularity being an asymptotically self-similar perturbation of a Burgers shock. The authors develop a modulation/self-similar framework, derive a closed bootstrap in weighted self-similar variables, and establish precise blowup time $T_*$, location $x_*$, and Hölder regularity $1/3$, along with convergence of the profile to a self-similar Burgers solution $\Psi_\nu$. The method hinges on a careful Lagrangian-trajectory analysis and a weighted transport framework to control both near-origin and far-field behavior, yielding a robust construction and stability result for the first shock. The work advances the understanding of singularity formation in nonlocal, nonlinear PDEs and delineates the regime where stable, self-similar shock formation persists under perturbations, with potential implications for broader nonlocal fluid models.

Abstract

For $0<α<\frac{1}{3}$ we construct unique solutions to the fractal Burgers equation $\partial_t u + u\partial_xu + (-Δ)^αu = 0$ which develop a first shock in finite time, starting from smooth generic initial data. This first singularity is an asymptotically self-similar, stable $H^6$ perturbation of a stable, self-similar Burgers shock profile. Furthermore, we are able to compute the spatio-temporal location and Hölder regularity for the first singularity. There are many results showing that gradient blowup occurs in finite time for the supercritical range, but the present result is the first example where singular solutions have been explicitly constructed and so precisely characterized.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 2.1
• Corollary 2.1.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 4.1
• Lemma 5.1
• Lemma 5.2