Recent Applications of Sobolev spaces part3(Machine Learning 2024)

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

Authors: Qionglei Chen, Yao Nie, Weikui Ye

Abstract: In this paper, we prove a sharp ill-posedness result for the incompressible non-resistive MHD equations. In any dimension d≥2, we show the ill-posedness of the non-resistive MHD equations in Hd2−1(Rd)×Hd2(Rd), which is sharp in view of the results of the local well-posedness in Hs−1(Rd)×Hs(Rd)(s>d2) established by Fefferman et al.(Arch. Ration. Mech. Anal., \textbf{223} (2), 677–691, 2017). Furthermore, we generalize the ill-posedness results from Hd2−1(Rd)×Hd2(Rd) to Besov spaces Bdp−1p,q(Rd)×Bdpp,q(Rd) and B˙dp−1p,q(Rd)×B˙dpp,q(Rd) for 1≤p≤∞,q>1. Different from the ill-posedness mechanism of the incompressible Navier-Stokes equations in B˙−1∞,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.