Shock Formation of the Burgers--Hilbert Equation

Source 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 spatial derivative of the solution blows up in finite time. 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 Hölder 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 which is compatible with the Hilbert transform, 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 the Hilbert transform and for transport equations.