Frontiers of Operator Theory / Frontières de la théorie des opérateurs

Collection Frontiers of Operator Theory / Frontières de la théorie des opérateurs

Organizer(s) Badea, Catalin ; Bayart, Frédéric ; Gallardo-Gutiérrez, Eva A. ; Grivaux, Sophie ; Lefèvre, Pascal
Date(s) 11/29/21 - 12/3/21
linked URL https://conferences.cirm-math.fr/2388.html
00:00:00 / 00:00:00
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Patrick Gérard and I introduced the cubic Szegö equation around ten years ago as a toy model of a totally non dispersive degenerate Hamiltonian equation. Despite of the fact that it is a complete integrable system, we proved that this equation develops some cascades phenomena. Namely, for a dense set of smooth initial data, the Szegö solutions have unbounded high Sobolev trajectories, detecting transfer of energy from low to high frequencies. However, this dense set has empty interior and a lot of questions remain opened to understand turbulence phenomena. Among others, we would like to understand how interactions of Fourier coefficients interfere on it. In a recent work, Biasi and Evnin explore the phenomenon of turbulence on a one parameter family of equations which goes from the cubic Szegö equation to what they call the 'truncated Szegö equation'. In this latter, most of the Fourier mode couplings are eliminated. However, they prove the existence of unbounded trajectories for simple rational initial data. In this talk, I will explain how, paradoxically, the turbulence phenomena may be promoted by adding a damping term. Those results are closely related to an inverse spectral theorem we proved on the Hankel operators.

Information about the video

Citation data

  • DOI 10.24350/CIRM.V.19855703
  • Cite this video Grellier Sandrine (11/29/21). Turbulent cascades for a family of damped Szegö equations. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19855703
  • URL https://dx.doi.org/10.24350/CIRM.V.19855703

Bibliography

  • GERARD, Patrick et GRELLIER, Sandrine. On a damped Szego equation (with an appendix in collaboration with Christian Klein). SIAM Journal on Mathematical Analysis, 2020, vol. 52, no 5, p. 4391-4420. - https://doi.org/10.1137/19M1299189
  • GÉRARD, Patrick, GRELLIER, Sandrine, et HE, Zihui. Turbulent cascades for a family of damped Szeg\" o equations. arXiv preprint arXiv:2111.05247, 2021. - https://arxiv.org/abs/2111.05247

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