Lowest order Carleman linearization for low Reynolds long-term behaviour of fluid flow simulations

It is shown that the lowest (second) order truncation of the Carleman linearization of the fluid equations (C2) recovers the late stage of the evolution, namely the steady-state solution, although to a decreasing degree of accuracy at increasing Reynolds number. This asymptotic property is first proved analytically for the decaying logistic with external forcing and then shown to hold to a significant degree of accuracy also for the more complex case of two-dimensional Kolmogorov-like fluid flow at low Reynolds numbers, below $Re\sim 10$. This time-asymptotic property may open interesting prospects for the quantum simulation of low-Reynolds steady-state fluid flows.