Nonequilibrium ensembles for the three-dimensional Navier-Stokes equations 

Abstract

At the molecular level fluid motions are, by first principles, described by time reversible laws. On the other hand, the coarse grained macroscopic evolution is suitably described by the Navier-Stokes equations, which are inherently irreversible, due to the dissipation term. Here, a reversible version of three-dimensional Navier-Stokes is studied, by introducing a fluctuating viscosity constructed in such a way that enstrophy is conserved, along the lines of the paradigm of microcanonical versus canonical treatment in equilibrium statistical mechanics. Through systematic simulations we attack two important questions: (a) What are the conditions that must be satisfied in order to have a statistical equivalence between the two non-equilibrium ensembles? (b) What is the empirical distribution of the fluctuating viscosity observed by changing the Reynolds number and the number of modes used in the discretization of the evolution equation? The latter point is important also to establish regularity conditions for the reversible equations. We find that the probability to observe negative values of the fluctuating viscosity becomes very quickly extremely small when increasing the effective Reynolds number of the flow in the fully resolved hydro dynamical regime, at difference from what was observed previously. a

Navier-Stokes and equivalence conjectures

“…(2) the simulations with large N show that α(u(t)) is observed, after a transient, not only > 0 but also quite close to 1: in [4] evidence is provided that this might be an illusion: following the evolution of α(t) on typical RN S N solutions it is found that as N increases the negative values of α have a rapidly decreasing probability. So the negative values of α might be not observable within the precision of the simulation and the time available to it.…”

“…Conjecture 3 is introduced in [4] to cover at least the results of the corresponding simulations: the simulations were not developed enough to allow stating that the failure of conjecture 2 on observables of scales over ≃ 1 8 K ν , as apparently shewed by the simulations, could be firmly confirmed; the point was left for consideration in future work.…”

“…This kinematic inequality (i.e. depending on u ∈ C ∞ and unrelated to the RN S N ) is obtained by combining the Hölder and Sobolev's inequalities, see for instance [4…”

“…There are very few rigorous results, stressed in [3,4], supporting the conjectures. And some breakthrough results, [5,6], on the classical NS equation can be directly applied to the reversible equations.…”

“…Reversible equations conjectured to be equivalent for the purpose of modeling stationary states, under large scale forcing, of incompressible Navier-Stokes evolutions have been introduced since the 90's, [1,2]. Equivalence has been first conjectured to be asymptotic at vanishing viscosity, then at all viscosity for observables of large scale, [3], or just for observables of scale up to ∼Kolmogorov's scale, [4].…”