Unstable shock formation of the Burgers-Hilbert equation

2022/01/11 by Yang, Ruoxuan · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2201.04208

Abstract

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 H⁹-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 Hölder 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.

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• Shock formation in 1D conservation laws I: Inviscid structure

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