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 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 with the property that these Gibbs measures are invariant for this flow. Moreover, we prove that the deterministic inviscid shell model has a stationary solution with respect to these measures.