Monday, 07 December 2020

In this talk we consider the pressureless Euler system in dimension greater than or equal to two. Several works have been devoted to the search of solutions which satisfy the following adhesion or sticky particle principle: if two particles of the fluid do not interact, then they move freely keeping constant velocity, otherwise they join with velocity given by the balance of momentum. In dimension one the problem is well studied and sticky particle solutions for general initial data have been proved to exist and be unique. In general dimension, the existence and uniqueness of a sticky particle solution is certainly given if the initial data consists of a finite number of particles. However, in dimensions greater than or equal to two, A. Bressan and T. Nguyen proved that as soon as one allows initial data to be concentrated on infinitely many particles, one can build counterexamples to both existence and uniqueness of sticky particle solutions. Even the issue of giving a suitable notion of sticky particle solution for general initial data was largely open. In collaboration with S. Bianchini, we introduce a suitable lagrangian notion of sticky particle solution and we show that even though the Cauchy problem in this class is not well-posed for every measure-type initial data, there exists a comeager set of initial data in the weak topology giving rise to a unique sticky particle solution. Moreover, for any of these initial data the sticky particle solution is unique also in the larger class of dissipative solutions (where trajectories are allowed to cross) and is given by a trivial free flow concentrated on trajectories which do not intersect. In particular for such initial data there is only one dissipative solution and its dissipation is equal to zero. Thus, for a comeager set of initial data the problem of finding sticky particle solutions is wellposed, but the dynamics that one sees is trivial. Our notion of dissipative solution is stable under weak limits so as to include limits of energy dissipating approximation schemes.

Weak solutions to the Euler equations which can dissipate the energy were constructed mathematically by De Lellis, Szekelyhidi and collaborators. We numerically simulate one recent method of the construction by Buckmaster, et al. (2019). In this talk, we discuss some aspects of the mathematical construction and utility of the weak solutions in the light of the physics of turbulence. This is a joint work with Uriel Frisch and Laszlo Szekelyhidi.

Tuesday, 08 December 2020

We resurrect an old Heisenberg – Chandrasekhar technique and insight from randomly stirred models to study the crossover between Kolmogorov and Bolgiano-Obukhov scaling for fully developed turbulence in a stably stratified fluid. The crossover scales and their dependence on the Richardson number are clearly established.

TBA

Wednesday, 09 December 2020

We use a new measure of many-body chaos for classical systems---cross-correlators---to show that in a thermalised fluid (obtained from the Galerkin-truncated three-dimensional Euler and one-dimensional Burgers equations) characterised by a temperature $T$ and $N_G$ degrees of freedom, the Lyapunov exponent $\lambda$ scales as $N_G\sqrt{T}$. This scaling, obtained from detailed numerical simulations and theoretical estimates, provides compelling evidence not only for recent conjectures $\lambda \sim \sqrt{T}$ for chaotic, equilibrium, classical many-body systems, as well as, numerical results from frustrated spin systems, but also, remarkably, show that $\lambda$ scales linearly with the degrees of freedom in a finite-dimensional, classical, chaotic system.

I will first demonstrate the effects of spectral Fourier truncation on different conservative systems: the 1D inviscid Burgers equation, the 2 and 3D Euler equation and the 2 and 3D Gross-Pitaevskii equation (GPE). Then, I will exhibit how finite temperature effects in superfluids can be described by the truncated GPE and estimate the corresponding effective viscosity. Finally, I will show some recent results indicating that the truncated Euler dynamics can correctly reproduce the transitional dynamics in confined 2D turbulence and the energy spectrum in forced 3D turbulence, at scales larger than the forcing scale.

Thursday, 10 December 2020

The finite amount of dissipation of kinetic energy in turbulent fluid, where viscosity seemingly plays a vanishingly small role, is one of the main properties of turbulence, known as the dissipative anomaly. This property is grounded in the singular nature and deep irreversibility of turbulent flows, and plays a central role in our understanding of turbulence. We present new numerical measurements aiming at understanding how time irreversibility influences the relative motion of tracers transported by the flow. In particular, a strong interlink is found between the long-range, intermittent Lagrangian correlations of energy dissipation and violent separations between pairs of fluid particle trajectories.

The large-scale dynamics of 3D turbulence has been investigated since Leonardo da Vinci (1505) who wondered why vortices, generated at the pillars of a bridge in the Arno river (Florence), tended to be long-lived.

More than four centuries later, Kármán in 1938 and Kolmogorov in 1941 speculated that, in the limit of vanishing viscosity and in the absence of forces, the (mean) energy of 3D incompressible turbulence would asymptotically decline at long time as a power-law $E(t) \propto t^{-n} $. More recently, in 2019, the investigations of
Lászlό Székelyhidi and colleagues, using weak (distributional) solutions of the 3D incompressible Euler equations, showed that the total energy decay could be compatible with an almost arbitrary law. Three-dimensional simulations of strong solutions to the Navier-Stokes equations (Matsumoto, 2020) are beginning to give some evidence for non-power law decline of the energy.

What about the 1D Burgers equation? Phenomenology and simulations seemed, until recently, to indicate again a power-law decay. Very recently a rigorous proof was given in our group for this power-law decay under the condition that the initial potential is either the Brownian motion in space (or its fractional generalization introduced by Kolmogorov).

Very recently Frisch, Pandit, and Roy managed to extend this proof and the simulations and prove non-powerlaw decay for a composite Brownian motion, called here the large-scale multifractal Brownian motion. This seems to be the first case where multifractals appear at large scales.

Friday, 11 December 2020

The Navier-Stokes-Voigt (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the three-dimensional Navier-Stokes equations for the purpose of direct numerical simulations. In this talk I will derive several statistical properties of the invariant measures associated with the solutions of the three-dimensional NSV equations. Moreover, I will show that for fixed viscosity, ν > 0, as the regularizing parameter, α, tends to zero, there exists a subsequence of probability invariant measures converging, in a suitable sense, to a strong stationary statistical solution of the three-dimensional Navier-Stokes equations, which is a regularized version of the notion of stationary statistical solutions - a generalization of the concept of invariant measure introduced and investigated by Foias. This fact is also supported by direct numerical simulations, employing the NSV analogue of the Sabra shell model, for small values of the regularizing parameter α compared to the Kolmogorov length scale. However, for larger values of α simulations exhibit multiscaling inertial range, and the dissipation range is displaying low intermittency. These facts provide evidence that the NSV regularization may reduce the stiffness of direct numerical simulations of turbulent flows, with a small impact on the energy containing scales.

Several of my recent contributions , mostly with Edriss Titi and more recently with E. Wiedemann, A. and P. Gwiadza, were motivated by the following issues:

The role of boundary effect in mathematical theory of fluid mechanic and the similarity , in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As consequences:

• I will recall the Kolmogorov 1/3 law and the Onsager conjecture and compare them to the issue of anomalous energy dissipation.

• Give extensions : For general systems with an extra conservation laws about local and global conservation.

• Give several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations. Insisting that in such case the absence of anomalous energy dissipation is equivalent to the persistence of regularity in the zero viscosity limit.

In conclusion, acknowledging a series a breakthroughs, would compare the meaning of these results both between incompressible and compressible regimes and also in the presence of Kelvin Helmholtz vortex sheets.

Monday, 14 December 2020

Preliminary to the main topic, there has been an intriguing result of L. Arnold, H. Crauel and V. Wisthutz 1983 about stabilization by noise. M. Capinsly, during a talk by Arnold group in Bremen in 1987, conjectured that such result could have a counterpart for SPDEs of fluid mechanics and could lead to a result of improved mixing rate due to transport noise. Recently, Lucio Galeati, Dejun Luo and myself have proved results in this direction. The particular result of this talk is concerned with a difficult case: the heat loss through a boundary when the fluid is turbulent but zero at the boundary, being viscous. We idealize the fluid transport by a white noise in time with suitable space covariance and show that the decay of temperature is improved by the noise. Other related results, for mixing of Euler flows and delayed blow-up of certain PDE models, will be mentioned.

In this talk we plan to give a general overview of the KPZ problem. We shall then discuss a possible approach to the problem of universality based on global solutions to the random Hamilton-Jacobi equation. Next we introduce a geometrical renormalisation scheme based on properties of minimisers and shocks. Finally, we shall formulate several conjectures and present some rigorous results in their support.

Tuesday, 15 December 2020

We study the applicability of artificial intelligence tools to different problems in fluid dynamics, from the search of an optimal navigation strategy in complex environments to data reconstruction from partial measurements of turbulent flows. To solve navigation problems we follow the Reinforcement Learning approach. Here, we focus on finding the path that minimizes the navigation time between two given points in a fluid flow, known as the Zermelo’s problem [1]. Concerning data-assimilation, we explore the capability of Generative Adversarial Network (GAN) to generate missing data in turbulent configurations. In particular, we investigate on a quantitative basis, their use in reconstructing 2d damaged snapshots extracted from a large database of numerical configurations of 3d turbulence in the presence of rotation, a case with multi-scale random features where both large-scale organized structures and small-scale highly intermittent and non- Gaussian fluctuations are present [2,3].

**References**

[1] Biferale, L., Bonaccorso, F., Buzzicotti, M., Di Leoni, P. C., & Gustavsson, K. (2019). Zermelo’s problem: Optimal point-to-point navigation in 2D turbulent flows using Reinforcement Learning. Chaos: An Interdisciplinary Journal of Nonlinear Science 29.10 (2019): 103138.

[2] Buzzicotti, M., Bonaccorso, F., Di Leoni, P. C., & Biferale, L. (2020). Reconstruction of turbulent data with deep generative models for semantic inpainting from TURB-Rot database. arXiv preprint arXiv:2006.09179 (in press Physycal Review FLuids)

[3] Biferale, L., Bonaccorso, F., Buzzicotti, M. and di Leoni, P.C., 2020. TURB-Rot. A large database of 3d and 2d snapshots from turbulent rotating flows. arXiv:2006.07469.

I will explain how the non-commutativity of an evolution operator with spatiotemporal symmetries, namely temporal scalings and Galilean transformations, leads to the existence of a hidden scaling symmetry. This symmetry appears in a normalized representation, which includes a state-dependent change of time and transforms the intermittent statistics to a scale-invariant form. In particular, one obtains scaling exponents of structure functions as eigenvalues in terms of a self-similar normalized probability measure. This presentation follows the preprint arXiv:2010.13089.

Wednesday, 16 December 2020

Visual manifestations of intermittency in computations of three-dimensional Navier-Stokes fluid turbulence appear as the familiar low-dimensional or `thin' filamentary sets on which vorticity and strain accumulate as energy cascades down to small scales. The first task of this talk is to investigate how weak solutions of the Navier-Stokes equations can be associated with a cascade and, as a consequence, with an infinite sequence of inverse length scales. It turns out that this sequence converges to a finite limit. The second task is to show how these results scale with integer dimension D=1, 2 & 3. In the light of the occurrence of thin sets, I discuss the mechanism of how the fluid might find the smoothest, most dissipative class of solutions rather than the most singular.

Shell models have found wide application in the study of hydrodynamic turbulence because they are easily solved numerically even at very large Reynolds numbers. Although bereft of spatial variation, they accurately reproduce the main statistical properties of fully-developed homogeneous and isotropic turbulence. Moreover, they enjoy regularity properties which still remain open for the three-dimensional (3D) Navier-Stokes equations (NSEs). The goal of this study is to make a rigorous comparison between shell models and the NSEs. It turns out that only the estimate of the mean energy dissipation rate is the same in both systems. The estimates of the velocity and its higher-order derivatives display a weaker Reynolds number dependence for shell models than for the 3D NSEs. Indeed, the velocity-derivative estimates for shell models are found to be equivalent to those corresponding to a velocity gradient averaged version of the 3D Navier-Stokes equations (VGA-NSEs), while the velocity estimates are even milder. Numerical simulations over a wide range of Reynolds numbers confirm the estimates for shell models.

Thursday, 17 December 2020

Weak solutions of the incompressible Euler equations which are weak limits of vanishing viscosity Navier-Stokes solutions inherit, in two dimensions, conservation properties which are not available for general weak solutions. Research has focused on the behavior of energy and the $p$-moments of vorticity, always in fluid domains with no boundary. In this talk I will report on recent work in this direction

In 1949 Onsager conjectured the existence of Hoelder continuous solutions of the Euler equations which do not conserve the kinetic energy. A rigorous proof of his statement has been given by Isett in 2017, crowning a decade of efforts in the subject. Onsager's original statement is however motivated by anomalous dissipation in the Navier-Stokes equations: roughly speaking it would be desirable to show that at least some dissipative Euler flow is the ``vanishing viscosity limit''. As pointed out by Duchon and Robert, such limits need

to satisfy a local form of the energy dissipation, analogous to that of suitable weak solutions of the Navier-Stokes equations, pioneered by Scheffer and used by Caffarelly-Kohn-Nirenberg in their famous partial regularity paper. This lead Isett to consider a stronger version of the Onsager's conjecture. While the latter is still far from being solved, I will report on recent progress, joint work with Hyunju Kwon.

Friday, 18 December 2020

Incompressible fluids are described by the Navier-Stokes equations. Yet, it is not yet known whether the Cauchy problem is well posed for these equations, and whether solutions of finite energy are regular or unique.

From a practical point of view, two interesting questions arise: if there is a blow-up solution, what is his shape?

Is there a loss of unicity after the blow-up?

This talk describes experimental and numerical investigations that are devised to provide helpful intuitions to mathematicians regarding these two questions. In particular, I will show how to detect and characterize areas of “lesser regularity” and how to connect them with possible loss of unicity.

I will provide an introduction to randomly forced hydrodynamical equations, which play an important role in field-theoretical studies of the statistical properties of turbulence. I will then cover old results on the multiscaling of structure functions in these models. I will end with some recent results on such multiscaling in the randomly forced magnetohydrodynamical models. This recent work has been done in collaboration with Ganapati Sahoo, Nadia Bihari Padhan, Abhik Basu, Sadhitro De, Dipankar Roy, and Dhrubaditya Mitra.