Non-uniqueness of smooth solutions of the 5D magnetohydrodynamic equations from critical data

TL;DR

This paper extends recent findings on non-uniqueness of solutions from 3D Navier-Stokes to 5D magnetohydrodynamic equations, demonstrating multiple solutions from the same initial data in a higher-dimensional setting.

Contribution

It develops schemes inspired by prior work to show non-uniqueness of solutions for 5D MHD equations with initial data in BMO^{-1}.

Findings

Constructed two distinct global solutions from the same initial data.

Demonstrated non-uniqueness in 5D magnetohydrodynamic equations.

Extended non-uniqueness results to higher dimensions.

Abstract

Recently, Coiculescu and Palasek \cite{Coiculescu2025} shows the non-uniqueness of solutions for the 3D incompressible Navier-Stokes equations with initial data in $BMO^{-1}$. Inspired by their breakthrough work, we develop their schemes for the incompressible magnetohydrodynamic equations and obtain a similar result in 5 dimensional case. More precisely, we construct two distinct global solutions with a initial data, which has nonvanishing velocity and magnetic fields in $BMO^{-1}(\mathbb{T}^5)$.