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Ill-Posedness for the b-Family of Equations

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Abstract

For \(s<3/2\), it is shown that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces \(H^s\) on both the torus and the line when \(b>1\). The proof of ill-posedness depends on the value of b, where the most interesting case arises for \(b=3\), the Degasperis–Procesi equation. Considering that the b-family of equations 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 for \(b>1\).

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Acknowledgments

This work was partially supported by a grant from the Simons Foundation (#246116 to Alex Himonas). Also, the authors would like to thank the referees of the paper for constructive reports.

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

    A. Alexandrou Himonas, Katelyn Grayshan & Curtis Holliman

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  1. A. Alexandrou Himonas
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  2. Katelyn Grayshan
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  3. Curtis Holliman
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Correspondence to A. Alexandrou Himonas.

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Communicated by Tudor Stefan Ratiu.

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Himonas, A.A., Grayshan, K. & Holliman, C. Ill-Posedness for the b-Family of Equations. J Nonlinear Sci 26, 1175–1190 (2016). https://doi.org/10.1007/s00332-016-9302-0

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