## Invariance of the Gibbs measures for periodic generalized Korteweg-de Vries equations

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- by Andreia Chapouto and Nobu Kishimoto PDF
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## Abstract:

In this paper, we study the Gibbs measures for periodic generalized Korteweg-de Vries equations (gKdV) with quartic or higher nonlinearities. In order to bypass the analytical ill-posedness of the equation in the Sobolev support of the Gibbs measures, we establish deterministic well-posedness of the gauged gKdV equations within the framework of the Fourier-Lebesgue spaces. Our argument relies on bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Then, following Bourgain’s invariant measure argument, we construct almost sure global-in-time dynamics and show invariance of the Gibbs measures for the gauged equations. These results can be brought back to the ungauged side by inverting the gauge transformation and exploiting the invariance of the Gibbs measures under spatial translations. We thus complete the program initiated by Bourgain [Comm. Math. Phys. 166 (1994), pp 1–26] on the invariance of the Gibbs measures for periodic gKdV equations.## References

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## Additional Information

**Andreia Chapouto**- Affiliation: School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
- Address at time of publication: Department of Mathematics, University of California, Los Angeles, California 90095
- MR Author ID: 1190711
- Email: chapouto@math.ucla.edu
**Nobu Kishimoto**- Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
- MR Author ID: 841241
- Email: nobu@kurims.kyoto-u.ac.jp
- Received by editor(s): April 27, 2021
- Received by editor(s) in revised form: December 20, 2021
- Published electronically: September 28, 2022
- Additional Notes: The first author was supported by the Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. The second author was supported by JSPS (Grant-in-Aid for Young Researchers (B) no. 16K17626). The authors were also supported by the European Research Council (grant no. 637995 “ProbDynDispEq” and grant no. 864138 “SingStochDispDyn”).
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 8483-8528 - MSC (2020): Primary 35Q53
- DOI: https://doi.org/10.1090/tran/8699