Unstable shock formation of the Burgers-Hilbert equation

This paper proves the existence of unstable shocks of the Burgers-Hilbert equation conjectured in arXiv:2006.05568. More precisely, we construct smooth initial data with finite $H^9$-norm such that the solution in self-similar coordinates is asymptotic to the first unstable solution to the self-similar inviscid Burgers equation. The blowup profile is a cusp with H\"older 1/5 continuity with explicit blowup time and location. Unlike the previously established stable shocks, the initial data cannot be taken in an open set; instead, we control the two unstable directions by Newton's iteration.