On well-posedness of the Degasperis-Procesi equation
A. Alexandrou Himonas - Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States (email)
Abstract: 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: DP equation, Cauchy problem, Sobolev spaces,
well-posedness, non-uniform dependence on initial data,
approximate solutions, commutator estimate, conserved quantities.
Received: March 2010; Revised: February 2011; Available Online: June 2011.
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