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 equationThis 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 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.

Initial boundary value problems of the Degasperis-Procesi equation

“…[5,25,30,31,32,36,44,45,46,47] and the citations therein. For example, Yin proved local well-posedness of Eq.…”

Blow-Up Phenomena and Global Existence to a Weakly Dissipative Shallow Water Equation

“…After Eq. (1.4) appeared, it has attracted many researchers to discover its dynamics (see [29]- [36]). Yin [29,30] proved the local well-posedness of Eq.…”

“…(1.4) has not only global strong solutions [31,33], but also blow-up solutions [33]- [35]. Apart from these, Coclite and Karlsen [36] proved that it has global entropy weak solutions in For and under suitable mathematical transforms and several restrictions on its coefficients [37], Eq. (1.1) becomes (1.5) where a  0andb  0are arbitrary positive constants.…”

On the Cauchy problem for the Degasperis-Procesi equation

“…Analogous to the case of the Camassa-Holm equation [11,35], Henry [34] showed that smooth solutions to the Degasperis-Procesi equation have infinite speed of propagation (see also [49]). Coclite and Karlsen [9] also obtained global existence results for entropy weak solutions in the class of L 1 (R) ∩ BV (R) and the class of…”

“…Since the Degasperis-Procesi equation was derived, many papers were devoted to its study; cf. [9,31,34,42,44,45,46,47,48,55] and the citations therein. For example, Yin proved local well-posedness to (1.5) with initial data u 0 ∈ H s , s > 3 2 on the line [55] and the precise blow-up scenario and a blow-up result were derived.…”