On well-posedness of the   Degasperis-Procesi equation

A. Alexandrou Himonas; Curtis Holliman

doi:10.3934/dcds.2011.31.469

Article Contents

2011, Volume 31, Issue 2: 469-488. Doi: 10.3934/dcds.2011.31.469

This issue Previous Article Exponential attractors for lattice dynamical systems in weighted spaces Next Article Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions

On well-posedness of the Degasperis-Procesi equation

A. Alexandrou Himonas^1, and
Curtis Holliman^1,

1.
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556

Received: February 28, 2010

Revised: January 31, 2011

Published: May 2011

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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.

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Mathematics Subject Classification: Primary: 35Q53, 58F17.

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