On the wellposedness of the KdV equation on the space of pseudomeasures

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Abstract

In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudomeasures, also referred to as the Fourier Lebesgue space \(\mathscr {F}\ell ^{\infty }(\mathbb {T},\mathbb {R})\), where \(\mathscr {F}\ell ^{\infty }(\mathbb {T},\mathbb {R})\) is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space \(\mathscr {F}\ell ^{s,\infty }(\mathbb {T},\mathbb {R})\) with \(-1/2 < s \le 0\) and improves on a wellposedness result of Bourgain for small Borel measures as initial data. A key ingredient of the proof is a characterization for a distribution q in the Sobolev space \(H^{-1}(\mathbb {T},\mathbb {R})\) to be in \(\mathscr {F}\ell ^{\infty }(\mathbb {T},\mathbb {R})\) in terms of asymptotic behavior of spectral quantities of the Hill operator \(-\partial _{x}^{2} + q\). In addition, wellposedness results for the KdV equation on the Wiener algebra are proved.

Keywords

KdV equation Well-posedness Birkhoff coordinates 

Mathematics Subject Classification

Primary 37K10 Secondary 35Q53 35D05 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Mathematics, University of ZurichZurichSwitzerland

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