On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation
The authors prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive Degasperis-Procesi equation \[ \partial_t u-\partial_{txx}^{3}u+4u\partial_x u=3\partial_{x}u\partial_{xx}^{2}u+u\partial_{xxx}^{3}u. \] In a recent paper, the same authors proved for this equation the existence and uniqueness of \(L^1\cap BV\) 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 Oleǐnik-type estimate and to prove the uniquess 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.
scientific article; zbMATH DE number 5132871