On a model for the Navier–Stokes equations using magnetization variables

Abstract

It is known that in a classical setting, the Navier-Stokes equations can be reformulated in terms of so-called magnetization variables w that satisfyand relate to the velocity u via a Leray projection u = Pw. We will prove the equivalence of these formulations in the setting of weak solutions that are also in L ∞ (0, T ; H 1/2 ) ∩ L 2 (0, T ; H 3/2 ) on the 3-dimensional torus.Our main focus is the proof of global well-posedness in H 1/2 for a new variant of (1), where Pw is replaced by w in the second nonlinear term:This is based on a maximum principle, analogous to a similar property of the Burgers equations.