On the well-posedness of the Degasperis-Procesi equation

By Giuseppe Maria Coclite, Kenneth Hvistendahl Karlsen. 2006

291 citations recorded by the source corpus.

We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis–Procesi equation (DP)∂tu-∂txx3u+4u∂xu=3∂xu∂xx2u+u∂xxx3u. 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∩BV, while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class L2∩L4. Finally, we extend our results to a class of generalized Degasperis–Procesi equations.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/946b80a337de0809c12567c18056da1bb7381d63

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Citations in the corpus

Equations with Peakon Traveling Wave Solutions (2013, background): “For related results about them, we refer the reader to [2, 5–15, 17–19, 21] and the references therein.”
Well-posedness, wave breaking and peakons for a modified μ-Camassa-Holm equation (2014, methodology): “Wave breaking phenomena and global existence of solutions for the Degasperis–Procesi equation were investigated in [6,16,18,19,34,35].”
Global weak solutions and blow-up structure for the Degasperis–Procesi equation (2006, background): “Coclite and Karlsen [5] also obtained global existence results for entropy solutions in L1(R)∩ BV(R) and in L2(R)∩L4(R).”
WELL-POSEDNESS OF HIGHER-ORDER CAMASSA-HOLM EQUATIONS (2009, background): “3), but its solutions are in general discontinuous, see [6] and the references cited therein.”
Well-posedness and Continuity Properties of the Fornberg-Whitham Equation in Besov Spaces (2016, methodology): “This was motivated by the works of Himonas and Holliman [14, 15, 16], Himonas and Kenig [17], Himonas, Kenig and Misio lek [18],”
The Degasperis-Procesi equation on the half-line (2012, background): “ for (1.1) were presented in [13], where the existence of peakon solutions was also established. An interesting aspect of (1.1) is the existence of weak solutions with a very low degree of regularity [6]. In particular, along with the peakons, equation (1.1) also admits ‘shock-peakons’ [21]. These a”
On the Blow-up Phenomena for the Degasperis–Procesi Equation (2010, background): “Coclite and Karlsen [4] obtained global existence results for entropy weak solutions of Eq.”