Abstract:We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis-Procesi equationu 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 KdV equation). We prove existence and $L1$ stability (uniqueness) results for entropy weak solutions belonging to the class $L1 \cap BV$, while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class $L2\cap L4$. Finally, we extend our results to a class of generalized Degasperis-Procesi equations._{t}- u_{txx}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}. (1)

**Paper:**- Available as PDF (288 Kbytes).
**Author(s):**- G. M. Coclite, <giusepc@math.uio.no>
- K. H. Karlsen <kennethk@math.uio.no>
**Publishing information:****Comments:****Submitted by:**- <giusepc@math.uio.no> May 27 2005.

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