On the Truth, and Limits, of a Full Equivalence $${\mathbf{p \cong \,v^2 }}$$ p ≅ v 2 in the Regularity Theory of the Navier–Stokes Equations: A Point of View

Abstract

The motivation at the origin of this note is the well known sufficient condition for regularity of solutions to the evolution Navier–Stokes equations, sometimes referred to in the literature as Ladyzhenskaya–Prodi–Serrin’s condition. Such a condition requires that the velocity field \(\,v\,\), alone, satisfies sufficiently strong integrability requirements in space–time. On the other hand, a relation like \(\,{p \cong \,|v|^2}\,\), with p pressure field, is loosely suggested by the Navier–Stokes equations themselves. In three papers published nearly 20 years ago we have considered this problem. The results obtained there immediately suggest new interesting questions. In this paper, we propose, and solve, some of them, while many other related problems remain still open.