On the well-posedness for the Ideal MHD equations in the Triebel-Lizorkin spaces

TL;DR

This paper establishes local well-posedness and blow-up criteria for the ideal magnetohydrodynamics (MHD) equations within Triebel-Lizorkin spaces, addressing gaps in previous proofs related to Euler equations.

Contribution

It introduces new well-posedness results for ideal MHD equations in Triebel-Lizorkin spaces and corrects a key step in the proof for incompressible Euler equations.

Findings

Proves local well-posedness in Triebel-Lizorkin spaces

Derives blow-up criteria for smooth solutions

Fills a gap in the proof for Euler equations

Abstract

In this paper, we prove the local well-posedness for the Ideal MHD equations in the Triebel-Lizorkin spaces and obtain blow-up criterion of smooth solutions. Specially, we fill a gap in a step of the proof of the local well-posedness part for the incompressible Euler equation in \cite{Chae1}.