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Growth bound and nonlinear smoothing for the periodic derivative nonlinear Schrödinger equation

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Abstract

A polynomial-in-time growth bound is established for global Sobolev \(H^s({\mathbb {T}})\) solutions to the derivative nonlinear Schrödinger equation on the circle with \(s>1\). These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution.

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Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Bradley Isom, Dionyssios Mantzavinos & Atanas Stefanov

  2. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, 35294, USA

    Atanas Stefanov

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  1. Bradley Isom
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  2. Dionyssios Mantzavinos
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  3. Atanas Stefanov
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Bradley Isom was partially supported by a graduate research assistantship from grant NSF-DMS 1908626. Dionyssios Mantzavinos acknowledges partial support from grant NSF-DMS 2206270. Atanas Stefanov acknowledges partial support from grant NSF-DMS 1908626. The authors are thankful to the anonymous referee for detailed remarks and suggestions that led to the improvement of the paper.

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Isom, B., Mantzavinos, D. & Stefanov, A. Growth bound and nonlinear smoothing for the periodic derivative nonlinear Schrödinger equation. Math. Ann. 388, 2289–2329 (2024). https://doi.org/10.1007/s00208-022-02492-8

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  • DOI: https://doi.org/10.1007/s00208-022-02492-8

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