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 is displayed in terms of labels \((n,\,m)\) beginning at \((1,\,1)\), which 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, the only difference being a factor of 2 in the exponent. This appears to be a generalization of the Prodi-Serrin conditions for \(n\geq 1\).