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

Thomas Kappeler ^[1] ; Jan Molnar ^[1]
1. [1] University of Zurich
  
  University of Zurich
  
  Zürich, Suiza
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 2, 2018, págs. 1479-1526
Idioma: inglés
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Resumen
- 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 F∞(T, R), where F∞(T, R)is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space Fs,∞(T, R) with −1/2 < s ≤ 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(T, R) to be in F∞(T, R) in terms of asymptotic behavior of spectral quantities of the Hill operator −∂2 x +q. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved.