Notes on continuous weak solutions to the ideal Hall equation

TL;DR

This paper constructs continuous weak solutions to the ideal Hall equation with prescribed energy profiles using convex integration, advancing understanding of energy conservation in Hall-MHD systems.

Contribution

It introduces a convex integration approach to construct solutions for the ideal Hall equation, a novel step towards Onsager's conjecture for Hall-MHD.

Findings

Constructed continuous weak solutions with prescribed energy profiles.

Paved the way for proving Onsager's conjecture in Hall-MHD.

Extended convex integration techniques to Hall equations.

Abstract

Using convex integration, we construct continuous weak solutions to the ideal Hall equation with a given energy profile. This is a first step in the direction of proving the Onsager's conjecture for the more general Hall-MHD equations.