Shock interactions for the Burgers-Hilbert Equation

TL;DR

This paper analyzes shock interactions in the Burgers-Hilbert equation, providing an asymptotic description of solutions near shock interactions, including a novel construction of piecewise smooth solutions with a single shock.

Contribution

It introduces a new asymptotic framework for understanding shock interactions and constructs solutions with specific regularity properties for general initial data.

Findings

Asymptotic description of solutions near shock interactions

Construction of piecewise smooth solutions with a single shock

Characterization of solution behavior with |x|ln|x| asymptotics

Abstract

This paper provides an asymptotic description of a solution to the Burgers-Hilbert equation in a neighborhood of a point where two shocks interact. The solution is obtained as the sum of a function with $H^2$ regularity away from the shocks plus a corrector term having an asymptotic behavior like |x|ln|x| close to each shock. A key step in the analysis is the construction of piecewise smooth solutions with a single shock for a general class of initial data.