Uniqueness for a Stochastic Inviscid Dyadic Model

For the deterministic dyadic model of turbulence, there are examples of initial conditions in $l^2$ which have more than one solution. The aim of this paper is to prove that uniqueness, for all $l^2$-initial conditions, is restored when a suitable multiplicative noise is introduced. The noise is formally energy preserving. Uniqueness is understood in the weak probabilistic sense.Comment: 13 pages, no figures. Submitted to the Proceedings of the American Mathematical Societ

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. As a consequence, global regular solutions cannot exist. After some reductions, the main tool is the escape bahavior at infinity of a certain birth and death process.Comment: Published in at http://dx.doi.org/10.1214/11-AAP768 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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. As a consequence, global regular solutions cannot exist. After some reductions, the main tool is the escape bahavior at infinity of a certain birth and death process.Comment: Published in at http://dx.doi.org/10.1214/11-AAP768 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

A dyadic model on a tree

We study an infinite system of non-linear 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 a 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

Positive and non-positive solutions for an inviscid dyadic model. Well-posedness and regularity

We improve regolarity and uniqueness results from the literature for the inviscid dyadic model. We show that positive dyadic is globally well-posed for every rate of growth $\beta$ of the scaling coefficients k_n = 2^{bn}. Some regularity results are proved for positive solutions, namely \sup_n n^{-a} k_n^{1/3} X_n(t) < \infty for a.e. t and \sup_n k_n^{1/3-1/(3b)} X_n(t) \leq C t^{-1/3}$for all$t$. Moreover it is shown that under very general hypothesis, solutions become positive after a finite time

The Impact of Spatial Heterogeneity in Land Use Practices and Aquifer Characteristics on Groundwater Conservation Policy Cost

Estimation of agricultural policy cost for a given level of groundwater conservation requires the establishment of an accurate baseline condition. This is especially critical when the benefits and cost of any conservation program are generally estimated relative to the status quo policy or baseline situation. An inaccurate baseline estimate will lead to poor estimates of potential water conservation savings and agricultural policy cost. Over a 60-year planning horizon per acre net present value is as much as 29.8% higher for a study area when aquifer characteristics are assumed to be homogenous and set to their average area value than when the heterogeneity in aquifer characteristics is explicitly modeled.Aquifer Modeling, Economics, Resource /Energy Economics and Policy,

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.Comment: 13 page