TIDE · Turbulence and Interactions with Dispersive Equations and Fourier Analysis
Horizon Europe — Marie Skłodowska-Curie Actions
- Duration
- 2024-09-01 → 2026-08-31
- EU contribution
- €165,313
- Participants
- 2
- Scheme
- HORIZON-TMA-MSCA-PF-EF
Lines connect the coordinator with its partners.
Results in brief
Turbulence and Interactions with Dispersive Equations and Fourier Analysis
Turbulence is one of the most important mechanisms that govern our environment, affecting air travel, prediction of ocean surface and weather forecasting. As put in relevance by the 2021 Nobel prize to Giorgio Parisi, approximate theoretical models and their consequent numeric simulation have been vital to control and predict those phenomena. Still, a rigorous understanding of the mechanisms that drive turbulence remains one of the main open problems in mathematical physics. In particular, the rigorous mathematical description of many central properties of the theory such as intermittency, multifractality and energy redistribution remains underdeveloped. On the other hand, dispersive PDEs and, in particular, the non-linear Schrödinger equation (NLS) are among the most important mathematical objects studied in Analysis, and incidentally, they are used to model fluid and wave turbulence. For instance, NLS has been used to derive a kinetic equation, systematically applied to model the ocean surface. With this action, I propose to advance in the theories of both turbulence and dispersive PDEs by further exploring the interaction between them, which will result in a better understanding of the mathematical structure of turbulence. I propose a Work Package for each direction of this interaction: Work Package 1: study the concepts of intermittency and multifractality from a rigorous analytic point of view via well-known dispersive models like NLS and the Vortex Filament Equation, which will contribute towards a better understanding of the mathematical theory of turbulence. Work Package 2: combine classical dispersive PDE and Fourier Analysis problems with probabilistic techniques arising from the study of turbulence, which has the potential of opening a whole new research line. In particular, making use of my previous experience, to study the almost everywhere pointwise convergence problem for NLS and for other dispersive equations from a probabilistic viewpoint.
Data: CORDIS, © European Union
Project objective
The turbulence of fluids and waves is one of the most important mechanisms that govern our environment, affecting air travel, prediction of ocean surface or weather forecasting. Even if simulations are used every day to predict turbulence, its accurate mathematical description is one of the biggest open problems in Mathematical Physics. On the other hand, dispersive PDEs are central objects in Mathematical Analysis and are among the most successful models for several physical phenomena. In this project I propose to advance in both theories, turbulence and dispersive PDEs, by studying their interaction. The project is divided in two Work Packages, each focused on one direction of such interaction:WP1) To advance in the rigorous mathematical treatment of intermittency and multifractality, central concepts in turbulence. The approach proposed is to identify them in the Vortex Filament Equation, the 1D Schrödinger map on the sphere and the non-linear Schrödinger equation (NLS), well-established models connected through the Hasimoto transformation and directly related to fluid and wave turbulence respectively. This is a novelty with respect to previous works, which mainly focus on abstract, often isolated mathematical objects. WP2) To study the pointwise convergence problem to the initial datum, one of the most important problems in dispersive PDEs and Fourier Analysis, from a novel probabilistic approach coming from fluid and wave turbulence. This is motivated by recent progress in wave turbulence, which has put forth the value of probabilistic techniques.To solve the problems, techniques from Fourier Analysis, PDEs and probability, complemented by number theory and geometry, will be required. For this I will have the support and training of two co-supervisors, experts in complementary areas in Harmonic Analysis and PDEs. Besides, the project will benefit from interdisciplinary collaboration with physician experts in turbulence.
Original text from CORDIS.
Participants
- BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS · BilbaoCoordinatorSpain
- MASSACHUSETTS INSTITUTE OF TECHNOLOGY · CambridgeUnited States
Links
- View on CORDIS
- DOI: 10.3030/101104250
- https://ec.europa.eu/research/participants/documents/downloadPublic?documentIds=080166e51b48dea2&appId=PPGMS
Data: CORDIS, © European Union