On wellposedness of the DegasperisProcesi 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 nonperiodic cases that the datatosolution map for the DegasperisProcesi (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 wellposedness 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 wellposedness 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,
wellposedness, nonuniform dependence on initial data,
approximate solutions, commutator estimate, conserved quantities.
Received: March 2010; Revised: February 2011; Available Online: June 2011. 
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