# 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/d*t* *X _{n}*=

*k*

_{n−1}

*X*

_{n−1}

^{2}−

*k*

_{n}X_{n}X_{n+1}

for *n* in the positive integers. The
components *X _{n}* are smooth functions of time
and

*k*=2

_{n}^{n}. These equations formally conserve energy (i.e. the

*l*

^{2}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 *X _{n}*=

*a*/(

_{n}*t*+

*t*

_{0}), with

*a*∈

*l*

^{2}.

[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 *k _{n}*=λ

^{n}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
Δ^{5/4} (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 2* ^{d}* 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

*X*.

_{n}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
ζ* _{p}*=

*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 *k _{n}* 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

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- 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]
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*Annales de l'Institut Henri Poincaré, Probabilités et Statistiques*52.3 (2016): 1217-1247. - [ABCFP2016]
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*Journal of Statistical Physics*144 (2011): article 1171.