Invariant Gibbs measures of the energy for shell models of turbulence: the inviscid and viscous cases

Abstract

Gaussian measures of Gibbsian type are associated with some shell model of 3D turbulence; they are constructed by means of the energy, a conserved quantity for the 3D inviscid and unforced shell model. We prove the existence of a unique global flow for a stochastic viscous shell model and of a global flow for the deterministic inviscid shell model, with the property that these Gibbs measures are invariant for these flows.

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.…”

“…Their studies have generated many important results. We refer, for instance, to [5,20,[22][23][24]27,28,[38][39][40] and references therein for the results obtained and the advances that have been made so far.…”

Invariant Measures of Gaussian Type for 2D Turbulence

“…But, they retain many important features of the true hydrodynamical models. Instead of dealing with complex valued unknowns we deal with the real and imaginary part of each component of the scalar velocity field (for the basic settings we follow [4]); this defines a sequence {u n } n with u n ∈ R 2 . For x = (x 1 , x 2 ) ∈ R 2 we set |x| 2 = x 2 1 + x 2 2 and the scalar product in R 2 is x · y = x 1 y 1 + x 2 y 2 .…”

“…We set k n = √ λ n . The bilinear term B is defined by means of the components B n = (B n,1 , B n,2 ) as follows (see, e.g., [4]):…”

“…We point out that the Gaussian invariant measure that we consider here is not the Gibbs measure of the enstrophy considered for the 2D stochastic NavierStokes or deterministic 2D Euler equation in previous papers [1,2,4,13,16], but has a more regular support. In particular, the support of this measure is a Sobolev space of non negative exponent.…”