Norm Inflation and Ill-Posedness for the Degasperis-Procesi Equation |
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Authors: | A Alexandrou Himonas Curtis Holliman Katelyn Grayshan |
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Institution: | 1. Department of Mathematics , University of Notre Dame , Notre Dame , Indiana , USA himonas.1@nd.edu;3. Department of Mathematics , University of Notre Dame , Notre Dame , Indiana , USA |
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Abstract: | For s < 3/2, it is shown that the Cauchy problem for the Degasperis-Procesi equation (DP) is ill-posed in Sobolev spaces H s . If 1/2 ≤ s < 3/2, then ill-posedness is due to norm inflation. This means that there exist DP solutions who are initially arbitrarily small and eventually arbitrarily large with respect to the H s norm, in an arbitrarily short time. Since DP solutions conserve a quantity equivalent to the L 2-norm, there is no norm inflation in H 0 for these solutions. In this case, ill-posedness is caused by failure of uniqueness. For all other s < 1/2, the situation is similar to H 0. Considering that DP is locally well-posed in H s for s > 3/2, this work establishes 3/2 as the critical index of well-posedness in Sobolev spaces. |
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Keywords: | Cauchy problem Conserved quantities Degasperis-Procesi equation Ill-posedness Norm inflation Peakon-antipeakon solutions Sobolev spaces Well-posedness |
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