On the weak solutions for the MHD systems with controllable total energy and cross helicity

TL;DR

This paper demonstrates the non-uniqueness of solutions to the 3D viscous and resistive MHD system with controlled total energy and cross helicity, extending previous ideal MHD results.

Contribution

It introduces a novel convex integration scheme with six types of flows to establish non-uniqueness in viscous and resistive MHD systems.

Findings

Non-uniqueness of solutions in $C([0,T];L^2)$ for viscous/resistive MHD.

Control of total energy and cross helicity over time.

Extension of ideal MHD non-uniqueness results to viscous/resistive case.

Abstract

In this paper, we prove the non-uniqueness of three-dimensional magneto-hydrodynamic (MHD) system in $C([0,T];L^2(\mathbb{T}^3))$ for any initial data in $H^{\bar{\beta}}(\mathbb{T}^3)$~($\bar{\beta}>0$), by exhibiting that the total energy and the cross helicity can be controlled in a given positive time interval. Our results extend the non-uniqueness results of the ideal MHD system to the viscous and resistive MHD system. Different from the ideal MHD system, the dissipative effect in the viscous and resistive MHD system prevents the nonlinear term from balancing the stress error $(\RR_q,\MM_q)$ as doing in \cite{2Beekie}. We introduce the box flows and construct the perturbation consisting in six different kinds of flows in convex integral scheme, which ensures that the iteration works and yields the non-uniqueness.