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 satisfy \begin{equation}\label{Abs_magform} \partial_tw + (\mathbb{P} w \cdot\nabla)w + (\nabla \mathbb{P} w)^\top w - Δw =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- Δw =0. \end{equation} This is based on a maximum principle, analogous to a similar property of the Burgers equations.

14 pages. To appear in Journal of Differential Equations

References in corpus (3)

• Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics
• Holder continuity for a drift-diffusion equation with pressure
• An Eulerian-Lagrangian Form for the Euler Equations in Sobolev Spaces

Cited by in corpus (1)

• An Eulerian-Lagrangian Form for the Euler Equations in Sobolev Spaces