Weak solutions to the Hall-MHD equations whose singular sets in time have Hausdorff dimension strictly less than 1

TL;DR

This paper constructs weak solutions to the 3D Hall-MHD equations that are smooth in time except on a set with Hausdorff dimension less than 1, demonstrating non-uniqueness in this context.

Contribution

It introduces a convex integration method to produce non-Leray-Hopf weak solutions with controlled singular sets for the Hall-MHD equations.

Findings

Existence of weak solutions with singular sets of Hausdorff dimension less than 1

Non-uniqueness of weak solutions in the specified regime

Application of convex integration to Hall-MHD equations

Abstract

In this paper, we focus on the three-dimensional hyper viscous and resistive Hall-MHD equations on the torus, where the viscous and resistive exponent $\alpha\in [\rho, 5/4)$ with a fixed constant $\rho\in (1,5/4)$. We prove the non-uniqueness of a class of weak solutions to the Hall-MHD equations, which have bounded kinetic energy and are smooth in time outside a set whose Hausdorff dimension strictly less than 1. The proof is based on the construction of the non-Leray-Hopf weak solutions via a convex integration scheme.