We prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive Degasperis-Procesi equation $$ \pt u-\ptxx u+4u\px u=3\px u\pxx u +u\pxxx u. $$ In a recent paper \cite{Coclite:2005cr}, we proved for this equation the existence and uniqueness of $L^1 \cap BV$ weak solutions satisfying an infinite family of Kru\v{z}kov-type entropy inequalities. The purpose of this paper is to replace the Kru\v{z}kov-type entropy inequalities by an Ole{\u\i}nik-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).

On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation / Coclite, Giuseppe Maria; Karlsen, K. H.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 234:1(2007), pp. 142-160. [10.1016/j.jde.2006.11.008]

On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation

COCLITE, Giuseppe Maria;
2007-01-01

Abstract

We prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive Degasperis-Procesi equation $$ \pt u-\ptxx u+4u\px u=3\px u\pxx u +u\pxxx u. $$ In a recent paper \cite{Coclite:2005cr}, we proved for this equation the existence and uniqueness of $L^1 \cap BV$ weak solutions satisfying an infinite family of Kru\v{z}kov-type entropy inequalities. The purpose of this paper is to replace the Kru\v{z}kov-type entropy inequalities by an Ole{\u\i}nik-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).
2007
On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation / Coclite, Giuseppe Maria; Karlsen, K. H.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 234:1(2007), pp. 142-160. [10.1016/j.jde.2006.11.008]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/93863
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