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 λ which bridges continuously between the two systems. That is, for λ = 1 one obtains the full nonlinear model, and the corresponding linear advection model is achieved for λ = 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 λ. Moreover, we show the existence of a finite-dimensional random attractor for each value of λ and establish the upper semicontinuity property of this random attractors, with respect to the parameter λ. 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.

Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

“…One can establish (66) with the same argument. The estimates (65) and (66) easily imply (3) and (7). Now we need to check that B(·, ·) satisfies (4).…”

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

“…[42] and [17]. It was shown in [14,Proposition 1] that the nonlinear term B(· , ·) for the GOY and Sabra shell models satisfies Assumption 2.2 with H = H. For more mathematical results related to shell models we refer to [3,5,6,14] and references therein.…”

Ergodicity of Stochastic Shell Models Driven by Pure Jump Noise

“…Due to these facts these models and their stochastic counterparts have been the subject of intensive numerical and analytical studies during the last two decades. We refer, for instance, to the works of Barbato et al [4], Bessaih et al [8], Constantin et al [15], and Ditlvesen [17] for more recent and detailed review of results related to the physical and mathematical theory of shell models. In recent years the mathematical analysis of stochastic partial differential equations driven by Lévy processes began to draw more and more attention.…”