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

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.