2024 Simons Collaboration on Wave Turbulence Annual Meeting

The 2024 annual meeting of the Collaboration on Wave Turbulence was held at the Simons Foundation on December 5th and 6th. A highlight of the meeting was Yu Deng’s presentation, which outlined
his collaborative work with the principal investigators and postdoctoral researchers on solving Hilbert’s sixth problem for particles and waves. This achievement represents a significant milestone, as the wave component of Hilbert’s sixth problem had been a central challenge of the wave turbulence proposal. Remarkably, the collaboration exceeded its original objectives, achieving a groundbreaking derivation of the Boltzmann kinetic equation for particle systems—an issue that had baffled mathematicians and mathematical physicists for over a century. This success was made possible by the close-knit teamwork fostered among Simons grant members, exemplifying the strength of collaborative research.

The annual meeting was preceded by a three-day workshop at the Courant Institute, which focused on related topics and showcased contributions from collaborators and postdoctoral researchers. This workshop included updates on a wide range of research directions, from theoretical advancements to innovative applications in geophysical and other systems. These presentations inspired detailed discussions and vibrant conversations, setting the tone for the annual meeting.

The meeting itself featured overview talks by senior members of the collaboration and invited guest speakers, addressing emerging and related topics in wave turbulence. The diverse talks covered areas such as integrable turbulence, odd turbulence, rotating turbulence, and internal wave turbulence. Additionally, the workshop highlighted advancements in dynamical systems approaches to dispersive equations, including new techniques using KAM theory to study periodic and almost periodic solutions, as well as invariant tori.

This two-part event underscored the collaboration’s substantial progress across multiple fronts, reaffirming its role as a leader in the field of wave turbulence and its broader applications.

Laurent Chevillard
École Normale Superiéure de Lyon

Formulation and Simulation of a Linear Dynamics Leading to a Loss of Regularity
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We will begin with some reminders on the phenomenology of three-dimensional fluid turbulence and the observational aspects of the statistical behavior of solutions of the Navier-Stokes equations, forced by a smooth term in space. To account for the cascade phenomenon, which asymptotically leads to a velocity field that is continuous yet highly irregular (of Hölderian type), we propose a linear model dynamics capable of generating such rough fields from entirely smooth ingredients over infinite time. This will involve a theoretical and numerical study of a transport phenomenon, not in physical space but in Fourier space, allowing energy to be transferred (or cascaded) across scales. A scheme based on finite spectral volumes will then provide a coherent numerical representation of such dynamics. This is joint work with G. Apolinario, G. Beck, C.-E. Bréhier, I. Gallagher, R. Grande, J.-C. Mourrat and W. Ruffenach.
 

Vincenzo Vitelli
University of Chicago

Odd Turbulence

Fully developed turbulence is a universal and scale-invariant chaotic state characterized by an energy cascade from large to small scales at which it is eventually arrested by dissipation. In this talk, we show how to harness turbulent cascades to generate patterns. Pattern formation entails a process of wavelength selection, which can usually be traced to the linear instability of a homogeneous state. By contrast, our fully nonlinear mechanism is triggered by the non-dissipative arrest of turbulent cascades: energy piles up at an intermediate scale, which is neither the system size nor the smallest scales at which it is usually dissipated. The tunable wavelength of these cascade-induced patterns can be set by a non-dissipative transport coefficient called odd viscosity, ubiquitous in chiral fluids ranging from bioactive to quantum systems. Odd viscosity acts as a scale-dependent Coriolis-like force and leads to two-dimensionalization of the flow and suppression of intermittency. These effects are caused by parity-breaking waves that give rise to a regime of (odd) wave turbulence at small scales. Apart from odd viscosity fluids, we discuss how cascade-induced patterns can arise in natural systems, including atmospheric flows, stellar plasma such as the solar wind, or the pulverization and coagulation of objects or droplets in which mass rather than energy cascades.
 

Léonie Canet
University Grenoble Alpes

Space-Time Dependence of Correlation Functions in Turbulence and Related Models
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Calculating the statistical properties of homogeneous and isotropic fully developed turbulence, particularly intermittency effects, from the Navier-Stokes equations remains an open issue. The theoretical challenge lies in the need to close an infinite hierarchy of coupled equations that determine correlation functions. The functional renormalization group (FRG) offers a promising tool to tackle this problem, enabling a controlled closure in the large wavenumber limit. I will demonstrate how it allows for analytical results on the space-time dependence of generic multi-point correlation functions of turbulent velocity in this regime. I will compare these predictions with available results from direct numerical simulations and experiments. In the second part, I will examine related simplified models of turbulence, such as Kraichnan’s model for passive scalar turbulence and the stochastic Burgers equation, showing how a universal behavior emerges in the temporal decay of correlation functions. I will discuss how this originates from underlying extended symmetries common to these models.
 

Alex Ionescu
Princeton University

On the Wave Turbulence Theory of 2D Gravity Waves
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I will discuss recent work on the rigorous study of wave turbulence in water wave systems. Wave turbulence has attracted significant attention in recent years, especially in the context of semilinear models, such as semilinear Schrödinger equations or multi-dimensional KdV-type equations. However, the scenario here is different, as the water wave equations are quasilinear, and the solutions cannot be obtained by iterating the Duhamel formula due to unavoidable derivative loss. I will discuss a new strategy to address this issue, combining deterministic energy estimates with a dispersive propagation of randomness argument. This is joint work with Yu Deng and Fabio Pusateri.
 

Anne-Sophie de Suzzoni
Ecole polytechnique

Propagation of Gaussianity and Application to the Incompressible Euler Dynamics
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In this talk, I will present a result on the propagation of Gaussianity under the flow of Hamiltonian equations. Namely, I will explain why solutions to a Hamiltonian equation whose initial data follows a Gaussian law still satisfy the Wick formula at latter times in the large box regime. This process is connected to the propagation of chaos in weak turbulence. However, to emphasize the fact that it has nothing to do with dispersion, we illustrate it with the incompressible Euler dynamics. The result is twofold: a first part lies at the lever of quasisolutions and is relatively general, and the second part only applies to the Euler dynamics. We will explain how to improve this result in the case of a dispersive equation.
 

Ching-Yao Lai
Stanford University

Machine-Precision Neural Networks for Multiscale Dynamics
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Deep-learning techniques are increasingly applied to scientific problems where the precision of networks is crucial. Despite being deemed universal function approximators, neural networks, in practice, struggle to reduce prediction errors below a certain threshold, even with large network sizes and extended training iterations. To address this issue, we developed multi-stage neural networks that divide the training process into different stages, with each stage utilizing a new network optimized to fit the residue from the previous stage. We demonstrate that the prediction error from multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine precision of double-floating point within a finite number of iterations. This advancement mitigates the longstanding accuracy limitations of neural network training and can be used to address the spectral bias in multiscale problems, including numerically finding blow-up solutions in fluid dynamics.
 

Vlad Vicol
New York University

Shock Formation and Maximal Hyperbolic Development in Multi-D Gas Dynamics
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We consider the Cauchy problem for the multi-dimensional compressible Euler equations, evolving from an open set of compressive and generic smooth initial data. We construct unique solutions to the Euler equations which are as smooth as the initial data, in the maximal spacetime set characterized by: at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time until reaching the Cauchy data prescribed along the initial time-slice. This spacetime is sometimes referred to as the “maximal globally hyperbolic development” (MGHD) of the given Cauchy data. We prove that the future temporal boundary of this spacetime region is a singular hypersurface, consisting of the union of three sets: first, a co-dimension-2 surface of “first singularities” called the pre-shock; second, a downstream co-dimension-1 surface emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream co-dimension-1 surface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach. In order to establish this result, we develop a new geometric framework for the description of the acoustic characteristic surfaces, and combine this with a new type of differentiated Riemann-type variables which are linear combinations of gradients of velocity/sound speed and the curvature of the fast acoustic characteristic surfaces. This is a joint work with Steve Shkoller (University of California at Davis)
 

Yu Deng
University of Chicago

The Hilbert Sixth Problem: Particles and Waves
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A major component of the Hilbert sixth problem concerns deriving macroscopic equations of motion, and the associated kinetic equations, from microscopic first principles. In the classical setting of Boltzmann’s kinetic theory, this corresponds to deriving the Boltzmann equation from particle systems governed by Newtonian dynamics. In the theory of wave turbulence, it corresponds to deriving the wave kinetic equation from nonlinear dispersive equations.

In this talk, I will present recent joint work with Zaher Hani and Xiao Ma, where we examine the hard sphere model in the particle setting and the cubic nonlinear Schrödinger equation in the wave setting. In both cases, we derive the corresponding kinetic equation for as long as the solution to this kinetic equation exists. This represents a crucial step toward resolving the Hilbert sixth problem.