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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.

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Authors and Affiliations

  1. Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057, Zurich, Switzerland

    Thomas Kappeler & Jan Molnar

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  1. Thomas Kappeler
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  2. Jan Molnar
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Correspondence to Thomas Kappeler.

Additional information

Thomas Kappeler: Partially supported by the Swiss National Science Foundation.

Jan Molnar: Partially supported by the Swiss National Science Foundation.

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Kappeler, T., Molnar, J. On the wellposedness of the KdV equation on the space of pseudomeasures. Sel. Math. New Ser. 24, 1479–1526 (2018). https://doi.org/10.1007/s00029-017-0347-1

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