Non-uniform Continuity for the MHD equations with only Magnetic Diffusion

In this paper, we prove the non-uniform continuity of the data-to-solution map for the incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion in Sobolev spaces $H^s({\mathbb{R}}^d)$ for all $s>0$ and $d=2,3$. Our results are first studies on the non-uniform continuity of the data-to-solution map for the resistive MHD equations. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields ${\mathbf{B}}_{\mathbf{0}}\in {\mathbb{R}}^d$, which reveal that the strong magnetic background fields may provide the stabilization effect but still preserve the analytical feature of non-uniform continuity of the data-to-solution map.