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Article Dans Une Revue Advances in Mathematics Année : 2012

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

Résumé

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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Dates et versions

hal-00593976 , version 1 (18-05-2011)
hal-00593976 , version 2 (03-06-2011)
hal-00593976 , version 3 (20-07-2011)

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