Sharp ill-posedness for the non-resistive MHD equations in Sobolev spaces

In this paper, we prove a sharp ill-posedness result for the incompressible non-resistive MHD equations. In any dimension $d\ge 2$, we show the ill-posedness of the non-resistive MHD equations in $H^{\frac{d}{2}-1}({\mathbb{R}}^d)\times H^{\frac{d}{2}}({\mathbb{R}}^d)$, which is sharp in view of the results of the local well-posedness in $H^{s-1}({\mathbb{R}}^d)\times H^s({\mathbb{R}}^d)(s>\frac{d}{2})$ established by Fefferman et al.(Arch. Ration. Mech. Anal., $\mathbf{\text{223}}$ (2), 677-691, 2017). Furthermore, we generalize the ill-posedness results from $H^{\frac{d}{2}-1}({\mathbb{R}}^d)\times H^{\frac{d}{2}}({\mathbb{R}}^d)$ to Besov spaces $B_{p,q}^{\frac{d}{p}-1}({\mathbb{R}}^d)\times B_{p,q}^{\frac{d}{p}}({\mathbb{R}}^d)$ and ${\dot{B}}_{p,q}^{\frac{d}{p}-1}({\mathbb{R}}^d)\times {\dot{B}}_{p,q}^{\frac{d}{p}}({\mathbb{R}}^d)$ for $1\le p\le \mathrm{\infty },q>1$. Different from the ill-posedness mechanism of the incompressible Navier-Stokes equations in ${\dot{B}}_{\mathrm{\infty },q}^{-1}$ \cite{B,W}, we construct an initial data such that the paraproduct terms (low-high frequency interaction) of the nonlinear term make the main contribution to the norm inflation of the magnetic field.