Local well posedness for KdV with data in a subspace of $H^{-1}$ and applications to illposedness theory for KdV and mKdV

Details Local well posedness for KdV with data in a subspace of $H^{-1}$ and applications to illposedness theory for KdV and mKdV Local well posedness for KdV with data in a subspace of $H^{-1}$ and applications to illposedness theory for KdV and mKdV

2009-12-13

arxiv.org/abs/0912.2480v4

Mathematics

Analysis of PDEs

35 pages (improved version)

Scientific paper

We prove the local well posedness for the KdV equation in the modulation space $M^{-1}_{2,1}(\mathbb{R})$. Our method is to substitute the dyadic decomposition by the uniform decomposition in the discrete Bourgain space. This wellposedness result enables us to show that the solution map is discontinuous at the origin with respect to the $H^{s} $-topology as soon as $s<-1$. Making use of the Miura transform we also deduce a discontinuity result for the $ H^s(\Real) $-topology, $s<0 $, for the solution map associated with the focussing and defocussing mKdV equations.

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