Remarks on the separation of Navier–Stokes flows

Recently, strong evidence has accumulated that some solutions to the Navier–Stokes equations in physically meaningful classes are not unique. The primary purpose of this paper is to establish necessary properties for the error of hypothetical non-unique Navier–Stokes flows under conditions motivated by the scaling of the equations. Our first set of results show that some scales are necessarily active—comparable in norm to the full error—as solutions separate. ‘Scale’ is interpreted in several ways, namely via algebraic bounds, the Fourier transform and discrete volume elements. These results include a new type of uniqueness criteria which is stated in terms of the error. The second result is a conditional predictability criteria for the separation of small perturbations. An implication is that the error necessarily activates at larger scales as flows de-correlate. The last result says that the error of the hypothetical non-unique Leray–Hopf solutions of Jia and Šverák locally grows in a self-similar fashion. Consequently, within the Leray–Hopf class, energy can hypothetically de-correlate at a rate which is faster than linear. This contrasts numerical work on predictability which identifies a linear rate. Our results suggest that this discrepancy may be explained by the fact that non-uniqueness might arise from perturbation around a singular flow.