Estimating the convex relaxation of the ideal magnetohydrodynamics equations

TL;DR

The paper tackles the problem of explicitly characterizing the Λ-convex relaxation of the ideal MHD equations, recasting the system in Elsässer variables and formulating a differential inclusion with a nonlinear constraint represented by $K_{r,s}$. It introduces the laminates framework to obtain a nontrivial lower bound on the Λ-convex hull and constructs a robust upper bound by proving the Λ-convex hull lies inside a carefully defined set $\overline{U}$, using a convexity argument on a functional $H_\gamma$. The main contributions are an explicit first-order laminate criterion and a nontrivial Λ-convex upper bound, advancing toward an explicit relaxation of ideal MHD and enabling quantitative insights into turbulent mean fields and instabilities. This framework provides a rigorous tool for modeling weak limits of weak MHD solutions and has potential implications for understanding turbulence and instabilities such as the Velikhov mechanism in magnetohydrodynamic contexts.

Abstract

We investigate the explicit convex relaxation of the ideal magnetohydrodynamics equations. We provide a non-trivial lower estimate on the lamination hull and an upper estimate on the $Λ$-convex hull, the latter providing inequalities which will be satisfied by weak limits of weak solution of the ideal MHD equations, which serve as a model of averaged turbulent magnetohydrodynamical flows.

Theorems & Definitions (16)

• Proposition 2.1
• Remark 2.2
• Corollary 2.3
• Proposition 2.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof