On well-posedness of the Degasperis-Procesi equation
A. Alexandrou Himonas Curtis Holliman
Discrete & Continuous Dynamical Systems - A 2011, 31(2): 469-488 doi: 10.3934/dcds.2011.31.469
It is shown in both the periodic and the non-periodic cases that the data-to-solution map for the Degasperis-Procesi (DP) equation is not a uniformly continuous map on bounded subsets of Sobolev spaces with exponent greater than 3/2. This shows that continuous dependence on initial data of solutions to the DP equation is sharp. The proof is based on well-posedness results and approximate solutions. It also exploits the fact that DP solutions conserve a quantity which is equivalent to the $L^2$ norm. Finally, it provides an outline of the local well-posedness proof including the key estimates for the size of the solution and for the solution's lifespan that are needed in the proof of the main result.
keywords: non-uniform dependence on initial data conserved quantities. DP equation well-posedness approximate solutions Sobolev spaces commutator estimate Cauchy problem

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