Abstract: We prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive Degasperis–Procesi equation

u_t−u_txx+4uu_x=3u_xu_xx+uu_xxx

In a recent paper \cite{Coclite:2005cr}, we proved for this equation the existence and uniqueness of $L1 \cap BV$ weak solutions satisfying an infinite family of Kružkov-type entropy inequalities. The purpose of this paper is to replace the Kružkov-type entropy inequalities by an Oleĭnik-type estimate and to prove uniqueness via a nonlocal adjoint problem. An implication is that a shock wave in an entropy weak solution to the Degasperis–Procesi equation is admissible only if it jumps down in value (like the inviscid Burgers equation).