Shock formation for the Burgers-Hilbert equation

TL;DR

This paper proves finite time shock formation in the Burgers-Hilbert equation, demonstrating explicit blowup profiles and conditions through self-similar analysis and precise estimates.

Contribution

It establishes the first rigorous proof of shock formation with explicit blowup profiles for the Burgers-Hilbert equation.

Findings

Finite time blowup of solutions with smooth initial data.

Explicit self-similar shock profile with a cusp of Hölder 1/3.

Description of blowup time and location via explicit ODEs.

Abstract

We prove finite time blowup of the Burgers-Hilbert equation. We construct smooth initial data with finite $H^5$-norm such that the $L^\infty$-norm of the spacial derivative of the solution blows up. The blowup is an asymptotic self-similar shock at one single point with an explicitly computable blowup profile. The blowup profile is a cusp with H\"older $1/3$ continuity. The blowup time and location are described in terms of explicit ODEs. Our proof uses a transformation to modulated self-similar variables, the quantitative properties of the stable self-similar solution to the inviscid Burgers equation, an $L^2$-estimate in self-similar variables, and pointwise estimates for Hilbert transform and for transport equations.