Shock interactions for the Burgers-Hilbert Equation

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.