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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

On well-posedness of the Degasperis-Procesi equation

Pages: 469 - 488, Volume 31, Issue 2, October 2011

doi:10.3934/dcds.2011.31.469       Abstract        References        Full Text (447.3K)       Related Articles

A. Alexandrou Himonas - Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States (email)
Curtis Holliman - 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.
Mathematics Subject Classification:  Primary: 35Q53, 58F17.

Received: March 2010;      Revised: February 2011;      Published: June 2011.

 References