Stochastic attractors for shell phenomenological models of turbulence

Abstract

Recently, it has been proposed that the Navier-Stokes equations and a relevant linear advection model have the same long-time statistical properties, in particular, they have the same scaling exponents of their structure functions. This assertion has been investigate rigorously in the context of certain nonlinear deterministic phenomenological shell model, the Sabra shell model, of turbulence and its corresponding linear advection counterpart model. This relationship has been established through a "homotopy-like" coefficient $\lambda$ which bridges continuously between the two systems. That is, for $\lambda =1$ one obtains the full nonlinear model, and the corresponding linear advection model is achieved for $\lambda =0$. In this paper, we investigate the validity of this assertion for certain stochastic phenomenological shell models of turbulence driven by an additive noise. We prove the continuous dependence of the solutions with respect to the parameter $\lambda$. Moreover, we show the existence of a finite-dimensional random attractor for each value of $\lambda$ and establish the upper semicontinuity property of this random attractors, with respect to the parameter $\lambda$. This property is proved by a pathwise argument. Our study aims toward the development of basic results and techniques that may contribute to the understanding of the relation between the long-time statistical properties of the nonlinear and linear models.

References in corpus (4)

• A Note on the Regularity of Inviscid Shell Model of Turbulence
• On the Anomalous Scaling Exponents in Nonlinear Models of Turbulence
• Sharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence
• Statistical Properties of Nonlinear Shell Models of Turbulence from Linear Advection Models: Rigorous Results

Cited by in corpus (6)

• Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces
• Statistical properties of stochastic 2D Navier-Stokes equations from linear models
• On Stochastic Shell Models of Turbulence
• Cutoff ergodicity bounds in Wasserstein distance for a viscous energy shell model with Lévy noise
• Ergodicity of stochastic shell models driven by pure jump noise
• Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type