Coexistence of chaotic and non-chaotic states in the two-dimensional Gauss–Navier–Stokes dynamics

Recently, Gallavotti proposed an Equivalence Conjecture in hydrodynamics, which states that forced-damped fluids can be equally well represented by means of the Navier–Stokes equations (NS) and by means of time reversible modifications of NS called Gauss–Navier–Stokes equations (GNS). This Equivalence Conjecture received numerical support in several recent papers concerning two-dimensional fluid mechanics. The corresponding results rely on the fact that the NS and GNS systems only have one attracting set. Performing similar two-dimensional simulations, we find that there are conditions to be met by the GNS system for this to be the case. In particular, increasing the Reynolds number, while keeping fixed the number of Fourier modes, leads to the coexistence of different attractors. This makes difficult a test of the Equivalence Conjecture, but constitutes a spurious effect due to the insufficient spectral resolution. With sufficiently fine spectral resolution, the steady states are unique and the Equivalence Conjecture can be conveniently established.