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

TL;DR

This paper establishes local well-posedness for the KdV equation in a specific modulation space and explores the ill-posedness of KdV and mKdV in certain Sobolev spaces, revealing discontinuities in the solution map.

Contribution

It introduces a new approach using uniform decomposition in Bourgain spaces to prove well-posedness in $M^{-1}_{2,1}$ and links this to ill-posedness results for KdV and mKdV.

Findings

Well-posedness of KdV in $M^{-1}_{2,1}$

Discontinuity of solution map in $H^{s}$ for $s<-1$

Discontinuity for mKdV in $H^{s}$ for $s<0$

Abstract

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.