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On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation. (English) Zbl 1133.35028

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.

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35L05 Wave equation
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L65 Hyperbolic conservation laws
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