On the existence of periodic solutions to the modified Korteweg–de Vries equation below \(H^{1/2}({\mathbb {T}})\)

Abstract

Existence and a priori estimates for real-valued periodic solutions to the modified Korteweg–de Vries equation with initial data in \(H^s\) are established for \(s>0\). The short-time Fourier restriction norm method is employed to overcome the derivative loss. Further, non-existence of solutions below \(L^2\) is proved conditional upon conjectured linear Strichartz estimates.

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Notes

  1. 1.

    For the actually more involved energy estimate, see Sect. 6.

  2. 2.

    Strictly speaking, we had to consider \(f_{m,k_i}\) or \(f_{m,k_i,j_i}\), respectively, tracking the additional dependence on m. Since all the estimates below are uniform in m, we choose to drop dependence on m for the sake of brevity.

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Acknowledgements

Financial support by the German Science Foundation (IRTG 2235) is gratefully acknowledged. I would like to thank the anonymous referee for a careful reading and commenting on an earlier manuscript, which significantly improved the presentation.

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Correspondence to Robert Schippa.

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Schippa, R. On the existence of periodic solutions to the modified Korteweg–de Vries equation below \(H^{1/2}({\mathbb {T}})\). J. Evol. Equ. 20, 725–776 (2020). https://doi.org/10.1007/s00028-019-00538-0

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Keywords

  • Dispersive equations
  • Modified Korteweg–de Vries equation
  • Existence of solutions
  • Short-time Fourier restriction norm method

Mathematics Subject Classification

  • 35Q53
  • 42B37