Non-uniqueness for the hypo-dissipative compressible 3D magnetohydrodynamic equations

Abstract

We consider the compressible 3D magnetohydrodynamic (MHD) equations under general pressure laws. For all hypo-viscosities $(-Δ)^{α_1}$ and hypo-resistivity $(-Δ)^{α_2}$ with $α_1,α_2\in(0,1)$, we prove the non-uniqueness of weak solutions to 3D MHD equations which reveals that there exist infinitely many weak solutions with the same initial data. Also, for the weak solutions in $C^{\widetildeβ}_{t,x}$ to the compressible ideal MHD, where $\widetildeβ>0$, we prove that they are the strong vanishing viscosity and resistivity limit of the weak solutions to the hypo-dissipative compressible MHD.