Weak and Strong Solutions of the 3D Navier–Stokes Equations and Their Relation to a Chessboard of Convergent Inverse Length Scales

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 3 D Navier–Stokes equations, on a periodic domain V = [ 0 , L ] 3 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 2 m ( V ) . The ( 1 , 1 ) position corresponds to the inverse Kolmogorov length R e 3 / 4 . These estimates ultimately converge to a finite limit, R e 3 , as n , m → ∞ , 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 generalization of the Prodi–Serrin conditions for n ≥ 1 .