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Preprint 2006-004

On the Uniqueness of Discontinuous Solutions to the Degasperis–Procesi Equation

Giuseppe Maria Coclite and Kenneth Hvistendahl Karlsen

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

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).

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Giuseppe Maria Coclite,
Kenneth Hvistendahl Karlsen,
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; 2006-02-06.