Estimating the convex relaxation of the ideal magnetohydrodynamics equations

TL;DR

This paper explores convex relaxation techniques for the ideal magnetohydrodynamics equations, providing bounds on the lamination and $\\Lambda$-convex hulls to better understand weak solutions and turbulence modeling.

Contribution

It introduces new estimates on the lamination and $\\Lambda$-convex hulls, advancing the mathematical understanding of weak solutions in ideal MHD.

Findings

Lower estimate on lamination hull

Upper estimate on $\Lambda$-convex hull

Inequalities for weak limits of solutions

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 $\Lambda$-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.