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Coclite, Giuseppe M.; Karlsen, Kenneth H.

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

J. Differ. Equations 234, No. 1, 142-160 (2007).
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.
Reviewer: Meng Wang (Hangzhou)

Cited in 27 Documents

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

Keywords:

shallow water equation; integrable equation; entropy condition; Kružkov-type entropy inequalities; Oleǐnik-type estimate; nonlocal adjoint problem
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\textit{G. M. Coclite} and \textit{K. H. Karlsen}, J. Differ. Equations 234, No. 1, 142--160 (2007; Zbl 1133.35028)
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References:

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