Stable and Unstable Shock Formation of the Burgers-Hilbert Equation

Abstract

The study of singularities has been an important part in the analysis of PDEs. One key type of singularities is shock. In many cases the shock has a self-similarity structure. Recently, the modulated self-similarity technique has achieved success in fluid dynamic equations. In this thesis, we apply this technique to establish finite time shock formation of the Burgers-Hilbert equation. The shocks are asymptotic selfsimilar at one single point. The shocks can be stable or unstable, both of which have an explicitly computable singularity profile, and the shock formation time and location are described by explicit ODEs. For the stable shock, the initial data are in an open set in the 𝐻⁵ -norm, and the shock profile is a cusp with Hölder 1/3 continuity. For the unstable shock, the initial data are in a co-dimension 2 subset of the 𝐻⁹ space, and the shock profile is of Hölder 1/5 continuity. Both cases utilize a transformation to appropriated self-similar coordinates, the quantitative properties of the corresponding self-similar solution to the inviscid Burgers’ equation, and transport estimates. In the case of unstable shock, we, in addition, control the two unstable directions by Newton’s iteration.