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On the well-posedness of the Degasperis-Procesi equation. (English) Zbl 1090.35142

We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis-Procesi equation

t u- txx 3 u+4u x u=3 x u xx 2 u+u xxx 3 u·( DP )

This equation can be regarded as a model for shallow water dynamics and its asymptotic accuracy is the same as for the Camassa-Holm equation (one order more accurate than the Korteweg-de Vries equation). We prove existence and L 1 stability (uniqueness) results for entropy weak solutions belonging to the class L 1 BV, while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class L 2 L 4 . Finally, we extend our results to a class of generalized Degasperis-Procesi equations.

35Q35PDEs in connection with fluid mechanics
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q58Other completely integrable PDE (MSC2000)
35L30Higher order hyperbolic equations, initial value problems