# Sharp ill-posedness results for the KdV and mKdV equations on the torus

Abstract : We establish a new a priori bound for $L^2$-bounded sequences of solutions to the mKdV equations on the torus. This first enable us to construct weak solutions in $L^2$ for this equation and to check that the "solutions" constructed by Kappeler and Topalov in the defocusing case satisfy the equation in some weak sense. In a second time, we prove that the solution-map associated with the mKdV and the KdV equation are discontinuous for the $H^s(\T)$ topology for respectively $s<0$ and $s<-1$. These last results are sharp.
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https://hal.archives-ouvertes.fr/hal-00593976
Contributor : Luc Molinet <>
Submitted on : Wednesday, July 20, 2011 - 10:32:06 PM
Last modification on : Thursday, March 5, 2020 - 5:32:49 PM
Document(s) archivé(s) le : Friday, October 21, 2011 - 2:23:45 AM

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KdVmKdVToreversion3.pdf
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### Identifiers

• HAL Id : hal-00593976, version 3
• ARXIV : 1105.3601

### Citation

Luc Molinet. Sharp ill-posedness results for the KdV and mKdV equations on the torus. Advances in Mathematics, Elsevier, 2012, 230 (4-6), pp.1895-1930. ⟨hal-00593976v3⟩

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