Shell models of turbulence, deterministic
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  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.
|||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  we prove uniqueness of the solution for a large class which comprises solutions with positive components.
|||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  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  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.
|||Smooth solutions for the dyadic model||Barbato, Morandin, Romito||Nonlinearity||2011|
Paper  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  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.
|||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  we prove Tao conjecture for the dyadic (which was not really required, since paper  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.
|||Global regularity for a logarithmically supercritical hyperdissipative dyadic equation||Barbato, Morandin, Romito||Dynamics of Partial Differential Equations||2014|
Paper  proves Tao conjecture for three-dimensional Navier-Stokes equations. The statements and proofs are quite technical, but the corresponding paper on dyadic  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.
|||Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system||Barbato, Morandin, Romito||Analysis and PDE||2014|
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  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.
|||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  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.
|||Structure function and fractal dissipation for an intermittent inviscid dyadic model||Bianchi, Morandin||-submitted-||2016|
Shell models of turbulence, stochastic
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  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.
|||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.
|||Uniqueness for a stochastic inviscid dyadic model||Barbato, Flandoli, Morandin||Proceedings of the American Mathematical Society||2010|
Paper  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.
|||Stochastic inviscid shell models: Well-posedness and anomalous dissipation||Barbato, Morandin||Nonlinearity||2013|
|||A large deviation principle for the free energy of random Gibbs measures with application to the REM||Fedrigo, Flandoli, Morandin||Annali di Matematica Pura ed Applicata||2007|
|||A resummed branching process representation for a class of nonlinear ODEs||Morandin||Electronic Communications in Probability||2005|
|||TAMGeS: A three-array method for genotyping of SNPs by a dual-colour approach||Cozza, Morandin, Galfrè et al.||BMC Genomics||2007|
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