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

Abstract

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., \textbf{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 \(B^{\frac{d}{p}-1}_{p, q}(\mathbb{R}^d)\times B^{\frac{d}{p}}_{p, q}(\mathbb{R}^d)\) and \(\dot B^{\frac{d}{p}-1}_{p, q}(\mathbb{R}^d)\times \dot B^{\frac{d}{p}}_{p, q}(\mathbb{R}^d)\) for \(1\le p\le\infty, q>1\). Different from the ill-posedness mechanism of the incompressible Navier-Stokes equations in \(\dot B^{-1}_{\infty, q}\) \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.