Papers on shell models

Deterministic models

One of the most studied shell models is the so-called dyadic which was introduced by Desnianskii and Novikov in 1974 and discovered again by Katz and Pavlovic in 2005. The equations are

d/dt Xn=kn−1Xn−12−knXnXn+1

for n in the positive integers. The components Xn are smooth functions of time and kn=2n. These equations formally conserve energy (i.e. the l2 norm), but this can be made rigorous only if the components decay fast enough and hence the solution lives in some regular space.

In [1] we found the first direct proof that for any initial condition, after some finite amount of time, anomalous dissipation occurs, by escaping of energy at infinity. Moreover we found a class of self-similar solutions of the form Xn=an/(t+t0), with a∈l2.

[1] Energy dissipation and self-similar solutions for an unforced inviscid dyadic model Barbato, Flandoli, Morandin Transactions of the American Mathematical Society 2011

For the dyadic model, uniqueness does not hold in general, with easy examples of non-uniqueness when the solution is regular but the components are negative. In [2] we prove uniqueness of the solution for a large class which comprises solutions with positive components.

[2] A theorem of uniqueness for an inviscid dyadic model Barbato, Flandoli, Morandin Comptes Rendus Mathematique 2010

Anomalous dissipation happens in the dyadic because energy moves to higher and higher components with exponential speed and hence it disappears at infinity in finite time. This is the energy cascade mechanism believed to hold for realistic inviscid fluid models (i.e. Euler equations). Shell models of turbulence should be tailored to create this phenomenon and indeed [1] clarified that this happens for the dyadic. The next relevant question is whether adding a dissipation term analogous to the Laplacian in the Navier-Stokes equations in enough to regularize the solutions. (This being the millenium prize problem when referred to three-dimensional Navier-Stokes.) In [3] we answered this question for the dyadic (with coefficients coherent with 3d N-S), showing that, upon rescaling the components, there exists a regular invariant region, thus yielding smoothness of the solutions of the viscous dyadic model.

[3] Smooth solutions for the dyadic model Barbato, Morandin, Romito Nonlinearity 2011

Paper [4] fills in some gaps in the understanding of the inviscid dyadic, in particular it generalizes uniqueness for coefficients kn=λn with λ>2. This requires a much subtler approach than what was done in [2] for λ≤2. Moreover it is shown that under very general hypothesis on the initial condition (positivity and finite energy requirements are dropped) there is one solution which is regular for all positive times.

[4] Positive and non-positive solutions for an inviscid dyadic model: Well-posedness and regularity Barbato, Morandin Nonlinear Differential Equations and Applications 2013

The next two papers come together as they are somewhat similar. They answer a question raised by Terence Tao in 2009. It is well known that when trying to prove well-posedness for the Navier-Stokes equations, the viscosity term has an order of derivative which is critical when d=2 and supercritical when d=3. This is one reason for which in dimension 2 one can prove well-posedness by standard arguments, while in dimension 3 it is not so. One can get well-posedness in dimension 3 by raising the order of the viscosity using Δ5/4 (this is the critical exponent), in place of the Laplacian. (These models are of course hyperdissipative with respect to the usual Navier-Stokes equations.)

Tao proved a well-posedness result for a logarithmically supercritical hyperdissipative Navier-Stokes model, hence lowering the exponent 5/4 by a logarithmical correction. At the same time he conjectured that the same should be true also for a larger logarithmical correction, but not more than that for a general model with energy cascade.

In [5] we prove Tao conjecture for the dyadic (which was not really required, since paper [3] has a stronger statement) but including a more general class of shell models, which extends to a particular vector model (also introduced by Tao) for which anomalous dissipation and blow-up are proved for physical dissipativity exponent.

[5] Global regularity for a logarithmically supercritical hyperdissipative dyadic equation Barbato, Morandin, Romito Dynamics of Partial Differential Equations 2014

Paper [6] proves Tao conjecture for three-dimensional Navier-Stokes equations. The statements and proofs are quite technical, but the corresponding paper on dyadic [5] can be used as a general guide for the same approach in a much simpler setting. It is interesting to notice that this result is accomplished introducing a new and realistic shell model, which bounds the dynamics of the true Euler equations, hence yielding a powerful tool to control the cascade of energy in Euler and Navier-Stokes.

[6] Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system Barbato, Morandin, Romito Analysis and PDE 2014

Tree-indexed models

Even though the most common derivation of shell models is per Fourier decomposition, the original paper by Katz and Pavlovic relied on a wavelet decomposition, that naturally gives a system of equations indexed by a regular tree with 2d children for every node. In this tree every node corresponds to a dyadic cube and to the wavelet components supported on that cube. The original dyadic equation was obtained by collapsing all the components of the nodes j of generation |j|=n in a single component Xn.

In [7] we recover the dyadic equation indexed by the tree and show that anomalous dissipation and blow-up occur as in the standard dyadic model. There is also an existence statement, while uniqueness is still an open problem.

[7] A dyadic model on a tree Barbato, Bianchi, Flandoli, Morandin Journal of Mathematical Physics 2013

A major feature of the tree dyadic model, the one that the standard dyadic could never enjoy, is that it could give rise to spatial intermittency, a phenomenon believed to happen in realistic turbulent fluids. Numerical experiments of 3d Euler show that, due to some possibly fractal concentration of energy, there should be significant deviations from Kolmogorv theory of turbulence K41. These deviations show up for example in the exponent of the structure function, which is the straight line ζp=p/3 according to K41, but should be a strictly concave function according to numerical evidence.

The classical dyadic gives the straight line prescribed by K41, but the tree dyadic has a chance to grasp a more complex picture. In fact, in [8] we make a first step in that direction. The model is a little more complex, in that the coefficients of the equations are allowed to vary a bit from node to node. On the other hand we simplify the study by adding a fixed forcing term and looking for constant solutions only.

We prove that for a large class of choices of the coefficients there exists a unique finite energy constant solution and that this solution exhibits a realistic (concave) curve for the exponent of the structure function and (anomalous) energy dissipation concentrated on a fractal set.

[8] Structure function and fractal dissipation for an intermittent inviscid dyadic model Bianchi, Morandin Communications in Mathematical Physics 2017

Stochastic models

One natural generalization of any shell model is to add a random perturbation. Some authors have proposed additive noise (see for example Romito) or a random forcing on the first component (Friedlander et al. and Andreis et al.) while we introduce a multiplicative noise perturbation that exchanges energy between adjacent components in a conservative way. The coefficients of these terms are the same kn of the non-linear term.

In [9] we prove existence of a bounded energy solution and anomalous dissipation. The idea is to use Girsanov transform to get rid of the nonlinearity and then study only the linear stochastic terms through the closed equation of the second moments. The latter is the Kolmogorov equation of a continuous-time Markov chain on the positive integers. This Markov chain is transient and its minimal solution is dishonest, in that mass disappears. As a consequence, the stochastic system has anomalous dissipation and this feature can be brought back to the original system through the Girsanov transform.

[9] Anomalous dissipation in a stochastic inviscid dyadic model Barbato, Flandoli, Morandin Annals of Applied Probability 2011

Uniqueness of the solution is technically difficult and hence is presented in a separate paper. We use again Girsanov transform and the associated Markov chain.

[10] Uniqueness for a stochastic inviscid dyadic model Barbato, Flandoli, Morandin Proceedings of the American Mathematical Society 2010

Paper [11] generalizes the arguments used for the dyadic to a very large class of shell models of turbulence. In particular there are two models which are commonly used in numerical simulations, because their behaviour shows intermittency and solitons. Their names are GOY (Gledzer, Okitani and Yamada) and Sabra (L'vov et al.). Well-posedness for the deterministic versions of these models is an open problem, but their stochastic versions (with conservative multiplicative noise) are included inside the general model we propose, and hence they are well posed problems but the solution exhibit anomalous dissipation.

[11] Stochastic inviscid shell models: Well-posedness and anomalous dissipation Barbato, Morandin Nonlinearity 2013

Papers [9], [10] and [11] are based on a stochastic linear model whose solutions are studied in L². Since this is an inviscid model, it is reasonable to expect some solutions to live in less regular spaces, like what happens for 3d Euler equations: this was the object of a study by Brzezniak et al. that introduced the concept of moderate solutions, for irregular initial conditions and proved their existence. In [12] we extended the study of the same model, showing that moderate solutions are in fact finite-energy solution, because anomalous dissipation immediately regularizes even very irregular initial conditions.

[12] Linear Stochastic Dyadic Model Bianchi, Morandin Journal of Statistical Physics 2021

Bibliography

[DN1974]
V.N. Desnianskii and E.A. Novikov. "Simulation of cascade processes in turbulent flows." Prikladnaia Matematika i Mekhanika 38 (1974): 507-513.
[KP2005]
Nets Katz, and Nataša Pavlovic. "Finite time blow-up for a dyadic model of the Euler equations." Transactions of the American Mathematical Society 357.2 (2005): 695-708.
[Tao2009]
Terence Tao. "Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation." Analysis & PDE 2.3 (2009): 361-366.
[Tao2016]
Terence Tao. "Finite time blowup for an averaged three-dimensional Navier-Stokes equation." Journal of the American Mathematical Society 29.3 (2016): 601-674.
[Rom2014]
Marco Romito. "Uniqueness and blow-up for a stochastic viscous dyadic model." Probability Theory and Related Fields 158.3-4 (2014): 895-924.
[FGV2016]
Susan Friedlander, Nathan Glatt-Holtz, and Vlad Vicol. "Inviscid limits for a stochastically forced shell model of turbulent flow." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 52.3 (2016): 1217-1247.
[ABCFP2016]
Luisa Andreis, et al. "Strong existence and uniqueness of the stationary distribution for a stochastic inviscid dyadic model." Nonlinearity 29.3 (2016): 1156.
[Gle1973]
E.B. Gledzer "System of hydrodynamic type admitting two quadratic integrals of motion." Soviet Physics Doklady. Vol. 18. 1973.
[YO1987]
Michio Yamada, and Koji Ohkitani. "Lyapunov spectrum of a chaotic model of three-dimensional turbulence." Journal of the Physical Society of Japan 56.12 (1987): 4210-4213.
[LPPPV1998]
V.S. L'vov, et al. "Improved shell model of turbulence." Physical Review E 58.2 (1998): 1811.
[BFNZ2011]
Z. Brzezniak, F. Flandoli, M. Neklyudov, B. Zegarlinski. "Conservative Interacting Particles System with Anomalous Rate of Ergodicity." Journal of Statistical Physics 144 (2011): article 1171.