Norm Inflation and Ill-Posedness for the Degasperis-Procesi Equation

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.

Keywords:

• Cauchy problem
• Conserved quantities
• Degasperis-Procesi equation
• Ill-posedness
• Norm inflation
• Peakon-antipeakon solutions
• Sobolev spaces
• Well-posedness

2010 Mathematics Subject Classfication:

• Primary: 35Q53, 37K10

Acknowledgements

The authors would like to thank John Derwent for constructive suggestions.