Estimates for the 2D Navier-Stokes equations: the effects of forcing

TL;DR

The paper addresses how forcing regularity affects dissipation and attractor size in the 2D Navier-Stokes equations and how to convert Grashof-based bounds to Reynolds-number form. Using both vorticity-based and velocity-based analyses, it extends Reynolds-number dependent bounds to dot-H^s forcings with -1 <= s < 2 and reveals a three-regime structure as the Sobolev regularity s varies. For 0 <= s < 2, energy dissipation tends to decrease with increasing Reynolds number while enstrophy dissipation can grow, and the attractor dimension increases; for s >= 2 the classical dual-cascade picture is recovered; for -1 <= s < 0 only energy-dissipation-type bounds are guaranteed with possible Reynolds-number growth. The results quantify the critical role of forcing regularity in 2D turbulence, linking mathematical bounds to turbulence phenomenology and offering parallels with RG analyses and fractal-forcing scenarios.

Abstract

Mathematical estimates for the Navier-Stokes equations are traditionally expressed in terms of the Grashof number, which is a dimensionless measure of the magnitude of the forcing and hence a control parameter of the system. However, experimental measurements and statistical theories of turbulence are based on the Reynolds number. Thus, a meaningful comparison between mathematical and physical results requires a conversion of the mathematical estimates to a Reynolds-dependent form. In two dimensions, this was achieved under the assumption that the second derivative of the forcing is square integrable. Nonetheless, numerical simulations have shown that the phenomenology of turbulence is sensitive to the degree of regularity of the forcing. Therefore, we extend the available estimates for the energy and enstrophy dissipation rates as well as the attractor dimension to forcings in the Sobolev space of order $s$; i.e. forcings whose Fourier coefficients decay with the wavenumber $k$ faster than $k^{-s-1}$. We consider the range $-1\leqslant s\leqslant 2$, where $s=2$ corresponds to the known estimates, and $s=-1$ is the smallest value of $s$ for which weak solutions are known to exist. The main result is the existence of three distinct regimes as a function of the regularity of the forcing.