Periodic Reporting for period 1 - TIDE (Turbulence and Interactions with Dispersive Equations and Fourier Analysis)
Période du rapport: 2024-09-01 au 2026-08-31
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.
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.
WP1 – Intermittency and multifractality in the evolution of vortex filaments.
With the goal of improving our rigorous mathematical understanding of the mechanisms that drive
turbulence, the main objective of this work package was to explore the multifractality and
intermittency in well-established models for the dynamics of vortex filaments, namely the Vortex
Filament Equation, the Schrödinger map and the non-linear Schrödinger equation. More precisely,
the WP was divided in two objectives:
Objective O1. To study the intermittency and multifractality of the trajectories of the tangent vectors to polygonal vortex filaments.
Progress:
Multifractality and intermittency of the rational trajectories of the polygonal vortex filaments. Fully completed, publications [P3,P4]
Multifractality and intermittency of the irrational trajectories of the polygonal vortex filaments. Limited progress.
Multifractality and intermittency of the trajectories of the tangent vectors to polygonal vortex filaments. Good progress, but no results yet due to early termination
Objective O2. To analyze the intermittency and multifractality arising in the cubic NLS with the Dirac comb as the initial datum.
Progress: Good progress, but no results yet due to early termination.
WP2 – Probabilistic approach to dispersive and Fourier problems.
Many ideas in the classical theory of turbulence are statistical and require probabilistic ingredients.
This had its effect in the study of dispersive equations, especially due to the work by Bourgain,
who launched the probabilistic study of dispersive PDE. The proposal of this WP was to apply
these probabilistic techniques in Carleson’s convergence problem to initial data. In this setting, two
specific objectives were proposed:
Objective O3. To decrease the sufficient Sobolev exponent for the convergence of the periodic NLS for generic data using the cutting-edge technique of Random Tensors.
Progress:
Quintic NLS in 2D: Good progress. No results yet due to early termination.
Other nonlinearities and higher dimensions. Limited progress due to early termination.
Objective O4. To explore the probabilistic convergence problem for the fractional Schrödinger equation.
Progress:
Periodic counterexamples for integer powers of the Laplacian. Fully completed, publication [P1].
Counterexamples for non-integer powers of the Laplacian. Limited progress due to early termination.
Positive results for probabilistic convergence of fractional Laplacian. Not started due to early termination.
Fractal counterexamples for the sequential convergence problem (not proposed in Part B but obtained while working on the project). Fully completed, publication [P2]
With the goal of improving our rigorous mathematical understanding of the mechanisms that drive
turbulence, the main objective of this work package was to explore the multifractality and
intermittency in well-established models for the dynamics of vortex filaments, namely the Vortex
Filament Equation, the Schrödinger map and the non-linear Schrödinger equation. More precisely,
the WP was divided in two objectives:
Objective O1. To study the intermittency and multifractality of the trajectories of the tangent vectors to polygonal vortex filaments.
Progress:
Multifractality and intermittency of the rational trajectories of the polygonal vortex filaments. Fully completed, publications [P3,P4]
Multifractality and intermittency of the irrational trajectories of the polygonal vortex filaments. Limited progress.
Multifractality and intermittency of the trajectories of the tangent vectors to polygonal vortex filaments. Good progress, but no results yet due to early termination
Objective O2. To analyze the intermittency and multifractality arising in the cubic NLS with the Dirac comb as the initial datum.
Progress: Good progress, but no results yet due to early termination.
WP2 – Probabilistic approach to dispersive and Fourier problems.
Many ideas in the classical theory of turbulence are statistical and require probabilistic ingredients.
This had its effect in the study of dispersive equations, especially due to the work by Bourgain,
who launched the probabilistic study of dispersive PDE. The proposal of this WP was to apply
these probabilistic techniques in Carleson’s convergence problem to initial data. In this setting, two
specific objectives were proposed:
Objective O3. To decrease the sufficient Sobolev exponent for the convergence of the periodic NLS for generic data using the cutting-edge technique of Random Tensors.
Progress:
Quintic NLS in 2D: Good progress. No results yet due to early termination.
Other nonlinearities and higher dimensions. Limited progress due to early termination.
Objective O4. To explore the probabilistic convergence problem for the fractional Schrödinger equation.
Progress:
Periodic counterexamples for integer powers of the Laplacian. Fully completed, publication [P1].
Counterexamples for non-integer powers of the Laplacian. Limited progress due to early termination.
Positive results for probabilistic convergence of fractional Laplacian. Not started due to early termination.
Fractal counterexamples for the sequential convergence problem (not proposed in Part B but obtained while working on the project). Fully completed, publication [P2]
[P1] D. Eceizabarrena, X. Yu. Uniform periodic counterexamples to Carleson's convergence
problem with polynomial symbols. Preprint. arXiv:2408.13935. Submitted for publication.
[P2] C.-H. Cho, D. Eceizabarrena. Bourgain's counterexample in the sequential convergence problem for
the Schrödinger equation. J. Fourier Anal. Appl. 31 (2025), 29. arXiv: 2403.07253.
[P3] V. Banica, D. Eceizabarrena, A. R. Nahmod and L. Vega. Multifractality and intermittency in
the limit evolution of polygonal vortex filaments. Math. Ann. 391 (2025), 2873-2899. arXiv:
2309.08114.
[P4] V. Banica, D. Eceizabarrena, A. R. Nahmod and L. Vega. Multifractality and polygonal vortex
filaments. Journées équations aux dérivées partielles (2024), Talk no. 1, 13p. arXiv: 2412.04926.
problem with polynomial symbols. Preprint. arXiv:2408.13935. Submitted for publication.
[P2] C.-H. Cho, D. Eceizabarrena. Bourgain's counterexample in the sequential convergence problem for
the Schrödinger equation. J. Fourier Anal. Appl. 31 (2025), 29. arXiv: 2403.07253.
[P3] V. Banica, D. Eceizabarrena, A. R. Nahmod and L. Vega. Multifractality and intermittency in
the limit evolution of polygonal vortex filaments. Math. Ann. 391 (2025), 2873-2899. arXiv:
2309.08114.
[P4] V. Banica, D. Eceizabarrena, A. R. Nahmod and L. Vega. Multifractality and polygonal vortex
filaments. Journées équations aux dérivées partielles (2024), Talk no. 1, 13p. arXiv: 2412.04926.