Weak and strong solutions of the $3D$ Navier-Stokes equations and their relation to a chessboard of convergent inverse length scales

Abstract

Using the scale invariance of the Navier-Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of the $3D$ Navier-Stokes equations, an infinite `chessboard' of estimates for these inverse length scales is displayed in terms of labels $(n,\,m)$ corresponding to $n$ derivatives of the velocity field in $L^{2m}$. The $(1,\,1)$ position corresponds to the inverse Kolmogorov length $Re^{3/4}$. These estimates ultimately converge to a finite limit, $Re^3$, as $n,\,m\to \infty$, although this limit is too large to lie within the physical validity of the equations for realistically large Reynolds numbers. Moreover, all the known time-averaged estimates for weak solutions can be rolled into one single estimate, labelled by $(n,\,m)$. In contrast, those required for strong solutions to exist can be written in another single estimate, also labelled by $(n,\,m)$, the only difference being a factor of 2 in the exponent. This appears to be a generalisation of the Prodi-Serrin conditions for $n\geq 1$.