Sharp and strong non-uniqueness for the magneto-hydrodynamic equations

TL;DR

This paper demonstrates sharp and strong non-uniqueness of weak solutions to the 3D magneto-hydrodynamic equations within certain function spaces, extending previous results and including solutions with non-trivial magnetic fields.

Contribution

It establishes the non-uniqueness of weak solutions in the critical function space range and constructs non-Leray-Hopf solutions with specific regularity properties.

Findings

Weak solutions are non-unique in L^p_tL^∞_x for 1≤p<2.

Non-Leray-Hopf solutions exist in L^p_tL^∞_x∩L^1_tC^{1-ε}.

Results extend non-uniqueness to solutions with non-trivial magnetic fields.

Abstract

In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the three-dimensional magneto-hydrodynamic (MHD) system. More precisely, we show that any weak solution $(v,b)\in L^p_tL^{\infty}_x$ is non-unique in $L^p_tL^{\infty}_x$ with $1\le p<2$, which reveals the strong non-uniqueness, and the sharpness in terms of the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint $(2, \infty)$. Moreover, for any $1\le p<2$ and $\epsilon>0$, we construct non-Leray-Hopf weak solutions in $L^p_tL^{\infty}_x\cap L^1_tC^{1-\epsilon}$. The results of Navier-Stokes equations in \cite{1Cheskidov} imply the sharp non-uniqueness of MHD system with trivial magnetic field $b$. Our result shows the non-uniqueness for any weak solution $(v,b)$ including non-trivial magnetic field $b$.