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) 29/11/2021 - 03/12/2021

linked URL https://conferences.cirm-math.fr/2388.html

00:00:00 / 00:00:00

3 5

Turbulent cascades for a family of damped Szegö equations

By Sandrine Grellier

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

Date of recording 29/11/2021
Date of publication 03/01/2022
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19855703
Cite this video Grellier, Sandrine (29/11/2021). 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

Domain(s)

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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