On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation

We prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive Degasperis-Procesi equation $$ \pt u-\ptxx u+4u\px u=3\px u\pxx u +u\pxxx u. $$ In a recent paper \cite{Coclite:2005cr}, we proved for this equation the existence and uniqueness of $L^1 \cap BV$ weak solutions satisfying an infinite family of Kru\v{z}kov-type entropy inequalities. The purpose of this paper is to replace the Kru\v{z}kov-type entropy inequalities by an Ole{\u\i}nik-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)