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


Kappeler, Thomas; Molnar, Jan (2018). On the wellposedness of the KdV equation on the space of pseudomeasures. Selecta Mathematica, 24(2):1479-1526.

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 Fℓ∞(T,R), where Fℓ∞(T,R) is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space Fℓs,∞(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 Fℓ∞(T,R) in terms of asymptotic behavior of spectral quantities of the Hill operator −∂2x+q. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved.

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 Fℓ∞(T,R), where Fℓ∞(T,R) is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space Fℓs,∞(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 Fℓ∞(T,R) in terms of asymptotic behavior of spectral quantities of the Hill operator −∂2x+q. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2018
Deposited On:28 Feb 2018 07:12
Last Modified:03 Apr 2018 01:04
Publisher:Springer
ISSN:1022-1824
OA Status:Closed
Publisher DOI:https://doi.org/10.1007/s00029-017-0347-1

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Content: Accepted Version
Language: English
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Size: 500kB
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Embargo till: 2018-07-05