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∂tu−∂txx3u+4u∂xu=3∂xu∂xx2u+u∂xxx3u. In a recent paper [G.M. Coclite, K.H. Karlsen, On the well-posedness of the Degasperis–Procesi equation, J. Funct. Anal. 233 (2006) 60–91], we proved for this equation the existence and uniqueness of L1∩BVL1∩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).