Preprint 2005-028

On the Well-Posedness of the Degasperis-Procesi Equation

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

u_t- u_txx+4uu_x =3u_xu_xx +uu_xxx.      (1) 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.

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.