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

It is known that in a classical setting, the Navier--Stokes equations can be reformulated in terms of so-called magnetization variables $w$ that satisfy \begin{equation}\label{Abs_magform} \partial_tw + (\mathbb{P} w \cdot\nabla)w + (\nabla \mathbb{P} w)^\top w - \Delta w =0, \end{equation} and relate to the velocity $u$ via a Leray projection $u=\mathbb{P} w$. We will prove the equivalence of these formulations in the setting of weak solutions that are also in $L^\infty(0,T;H^{1/2})\cap L^2(0,T;H^{3/2})$ on the 3-dimensional torus. Our main focus is the proof of global well-posedness in $H^{1/2}$ for a new variant of this system, where $\mathbb{P} w$ is replaced by $w$ in the second nonlinear term: \begin{equation}\label{Abs_Simplified} \partial_tw + (\mathbb{P} w \cdot\nabla)w + \frac{1}{2}\nabla|w|^2- \Delta w =0. \end{equation} This is based on a maximum principle, analogous to a similar property of the Burgers equations.