Non-uniqueness of weak solutions to the 3-D stationary MHD equations in Besov space with negative regularity index

Abstract

In this paper, we study the 3-D incompressible fractional stationary MHD equations on the torus ${\mathbb{T}}^3$. For fractional power indices ${\alpha }_1,{\alpha }_2>0,$ we prove that there exist infinite non-trivial stationary singular solutions of 3-D fractional MHD equations via convex integration. In particular, this result shows the non-uniqueness of weak solutions to the 3-D stationary MHD equations in some sub-critical Besov spaces.