Velocity–pressure correlation in Navier–Stokes flows and the problem of global regularity

Abstract

Incompressible fluid flows are characterised by high correlations between low pressure and high velocity and vorticity. The velocity–pressure correlation is an immediate consequence of fluid acceleration towards low-pressure regions. On the other hand, fluid converging to a low-pressure centre is driven sideways by a resistance due to incompressibility, giving rise to the formation of a strong vortex, hence the vorticity–pressure correlation. Meanwhile, the formation of such a vortex effectively shields the low-pressure centre from incoming energetic fluid. As a result, a local pressure minimum can usually be found at the centre of a vortex where the vorticity is greatest but the velocity is relatively low, hence the misalignment of local pressure minima and velocity maxima. For Navier–Stokes flows, this misalignment has profound implications for extreme momentum growth and maintenance of regularity. This study examines the role of the velocity–pressure correlation in the problem of Navier–Stokes global regularity. On the basis of estimates for flows locally satisfying the critical scaling of the Navier–Stokes system, a qualitative theory of this correlation is considered. The theory appears to be readily quantified, advanced and tested by theoretical, mathematical and numerical methods. Regularity criteria depending on the degree of the velocity–pressure correlation are presented and discussed in light of this theory. The result suggests that as long as global pressure minimum (or minima) and velocity maximum (or maxima) are mutually exclusive, then regularity is likely to persist. This is the first result that makes use of an explicit measure of the velocity–pressure correlation as a key factor in the maintenance of regularity or development of singularity.