Orateur : Yassine Tahraoui
Établissement : Scuola Normale Superiore, Pisa (Italie)
Dates : 2025-04-17 – 2025-04-17
Heures : 14:00 – 14:00
Lieu : Salle 0-1
Résumé :
Turbulence is one of the most fascinating problems from a physical and mathematical point of view. Despite the considerable progress made in this field, many questions remain open to this day. My talk is about the effects of small-scale turbulence on large-scale motion by using scaling limit, started a few years ago after the development of a new technique [2]. In [2], to some extent, we can consider it as a variant of diffusion approximation results in stochastic homogenization theory: a stochastic first order differential operator in Stratonovich form gives rise, in a suitable scaling limit, to a deterministic diffusion term.
Many works have been done in recent years using the scaling limit in both scalar and vector cases. The second one is characterized by the presence of stretching, which is by no means an incremental detail over the scalar case. Our aim in [1] is understanding the stretching mechanism of stochastic models of turbulence acting on a simple model of polymer. We consider a turbulent model that is white noise in time and activates frequencies in a shell N ≤ |k| ≤ 2N and investigate the scaling limit as N → ∞, under suitable intensity assumption, such that the stretching term has a finite limit covariance. The polymer density equation, initially an SPDE, converges weakly to a limit deterministic equation with a new term. Stationary solutions can be computed and show power law decay in the polymer length, which corresponds to the physical predictions.
The activities mentioned herein were performed in the framework of the project: EU-HORIZON EUROPE ERC-2021-ADG “Noise in Fluids” (NoisyFluid), no. 101053472.
References
[1] F. Flandoli and Y. Tahraoui. Stretching of Polymers and turbulence: Fokker Planck equation, special
stochastic scaling limit and stationary law. https://arxiv.org/abs/2410.00520 (2024)
[2] L. Galeati. On the convergence of stochastic transport equations to a deterministic parabolic one. Stoch
PDE: Anal Comp 8, 833–868 (2020)