David Barbato

Uniqueness for a Stochastic Inviscid Dyadic Model

AbstractFor the deterministic dyadic model of turbulence, there are exam-ples of initial conditions in l 2 which have more than one solution. Theaim of this paper is to prove that uniqueness, for all l 2 -initial condi-tions, is restored when a suitable multiplicative noise is introduced.The noise is formally energy preserving. Uniqueness is understood inthe weak probabilistic sense. 1 Introduction The infinite system of nonlinear differential equationsdX n (t)dt= k n−1 X 2n−1 (t) −k n X n (t)X n+1 (t), t ≥ 0 (1.1)X n (0) = x n for n ≥ 1, with coefficients k n > 0 for each n ≥ 1, X 0 (t) = 0 and k 0 = 0, isone of the simplest models which presumablyreflect some of the propertiesof3D Euler equations. At least, it is infinite dimensional, formally conservative(the energyP ∞n=1 X 2n (t) is formally constant), and quadratic. One of its‘pathologies’ is the lack of uniqueness of solutions, in the space l 2 of squaresummable sequences: when, for instance, k n = λ n with λ > 1, there areexamples of initial conditions x = (x

Heterogeneous large total CO2 abundance in the shallow magmatic system of Kilauea volcano, Hawaii

[1] Due to its very low solubility in silicate melts, CO 2 concentrations in melt inclusions (MIs) within crystals are commonly orders of magnitude less than the total concentration in the multiphase magma, strongly limiting the possibility to constrain CO 2 abundance based on the dissolved quantities. Here we develop a statistical method to process MI data, which allows analytical uncertainties to be taken into account together with the peculiar features of the local saturation surface. The method developed leads to retrieve total H 2 O and CO 2 concentrations in magma as well as the gas phase abundance at the time of magma crystallization. Application to a set of 29 high-resolution secondary ion mass spectrometry (SIMS) MI data from a single specimen of the 1842-1844 eruption of Kilauea, Hawaii, reveals the existence of heterogeneous total CO 2 abundance, and of at least 2-6 wt % total CO 2 in some magma batches, two orders of magnitude higher than the dissolved amounts and 30-50 times more abundant than the corresponding total H 2 O content. Heterogeneous total volatile concentrations are interpreted as due to a combination of degassing and gas flushing in magma subject to convective motion at shallow depth where P 1 wt % is likely to characterize the >30 km deep magma, not represented in the analyzed inclusions, from which a CO 2 -rich gas phase exsolves and decouples from the liquid.

Energy dissipation and self-similar solutions for an unforced inviscid dyadic model

A shell-type model of an inviscid fluid, previously considered in the literature, is investigated in absence of external force. Energy dissipation of positive solutions is proved, and decay of energy like t ―2 is established. Self-similar decaying positive solutions are introduced and proved to exist and classified. Coalescence and blow-up are obtained as a consequence, in the class of arbitrary sign solutions.

Smooth solutions for the dyadic model

We consider the dyadic model, which is a toy model to test issues of well-posedness and blow-up for the Navier-Stokes and Euler equations. We prove well-posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier-Stokes. Likewise we prove well-posedness for the inviscid problem (in a suitable regularity class) when the parameter corresponds to the strongest transport effect of the non-linearity.

Anomalous dissipation in a stochastic inviscid dyadic model

A stochastic version of an inviscid dyadic model of turbulence, with multiplicative noise, is proved to exhibit energy dissipation in spite of the formal energy conservation. The proof is based on the reduction to a linear stochastic equation with multiplicative noise, by Girsanov transform, and the interpretation of the second moment equation as the master equation of a birth and death process.

Global regularity for a slightly supercritical hyperdissipative Navier–Stokes system

We prove global existence of smooth solutions for a slightly supercritical hyperdissipative Navier--Stokes under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao [Tao2009].

On a stochastic Leray-α model of Euler equations

We deal with the 3D inviscid Leray-α model. The well posedness for this problem is not known; by adding a random perturbation we prove that there exists a unique (in law) global solution. The random forcing term formally preserves conservation of energy. The result holds for the initial velocity of finite energy and the solution has finite energy a.s. These results continue to hold in the 2D case.

A dyadic model on a tree

We study an infinite system of nonlinear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3D Euler and Navier-Stokes equations in a rough approximation of wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.

Stochastic inviscid shell models: well-posedness and anomalous dissipation

In this paper we study a stochastic version of an inviscid shell model of turbulence with multiplicative noise. The deterministic counterpart of this model is quite general and includes inviscid GOY and Sabra shell models of turbulence. We prove global weak existence and uniqueness of solutions for any finite energy initial condition. Moreover energy dissipation of the system is proved in spite of its formal energy conservation.

Strong existence and uniqueness of the stationary distribution for a stochastic inviscid dyadic model

We consider an inviscid stochastically forced dyadic model, where the additive noise acts only on the first component. We prove that a strong solution for this problem exists and is unique by means of uniform energy estimates. Moreover, we exploit these results to establish strong existence and uniqueness of the stationary distribution.