Uniqueness for an inviscid stochastic dyadic model on a tree

Abstract

In this paper we prove that the lack of uniqueness for solutions of the tree dyadic model of turbulence is overcome with the introduction of a suitable noise. The uniqueness is a weak probabilistic uniqueness for all $l^2$-initial conditions and is proven using a technique relying on the properties of the $q$-matrix associated to a continuous time Markov chain

2D Euler Equations with Stratonovich Transport Noise as a Large-Scale Stochastic Model Reduction

“…Precursors already appeared several years ago, see, for instance, (Brzeźniak et al 1991(Brzeźniak et al , 1992Rozovskii 2004, 2005). Then, it was observed, for particular models (see, for instance, Flandoli et al 2010;Maurelli 2011;Flandoli et al 2014;Barbato et al 2014;Bianchi 2013;Beck et al 2014; Flandoli 2010; Bianchi and Flandoli 2020 and others) that such noise has sometimes rich regularizing properties, typically in terms of improved uniqueness results or blow-up control. This also contributed to additional investigations on such random perturbation.…”

Linear Stochastic Dyadic Model

“…The model (1) considered in this paper is itself stochastic, through the random initial condition X , and the infinite-dimensional multiplicative noise (W n ) n , which formally conserves energy. Other examples linked to this one in the literature are the already mentioned [7][8][9]11] and [14].…”

“…The requirement of finite fourth moments, which appears in the definition of proper solution, is a technical assumption needed in Proposition 1, which shows that second moments of a proper solution solve a closed system of equations (see also [9], [11]). Fourth moments play also a role in Theorem 1.…”

“…To prove the bound (11) of the H −1 norm, we can notice that Lemma 1 applies to the approximants X (N ) and, taking the limit, the same inequality holds for X . Then it is enough to recall that when σ > 0 we need to take the auxiliary proper solution Z into account, for which (4) applies, so that, with the H −1 norm controlled by the l 2 one,…”

“…It is closely related to dyadic models of turbulence, an interesting simplification of the energy cascade phenomenon, which have been extensively studied in the physical literature, as well as the mathematical one. We mention here just some results: in [7][8][9]11] one can find Communicated by Michael Kiessling.…”