Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces

TL;DR

This paper proves local existence and uniqueness of solutions for the non-resistive MHD equations in nearly optimal Sobolev spaces, using maximal regularity estimates and addressing the limitations of initial data regularity.

Contribution

It establishes local well-posedness for the non-resistive MHD equations in Sobolev spaces with minimal regularity assumptions, extending previous results.

Findings

Proves local existence and uniqueness in Sobolev spaces for d=2,3.

Identifies the regularity threshold for initial data.

Provides an explicit example illustrating the failure of certain regularity properties.

Abstract

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $\mathbb{R}^d$, $d=2,3$, with initial data $B_0\in H^s(\mathbb{R}^d)$ and $u_0\in H^{s-1+\varepsilon}(\mathbb{R}^d)$ for $s>d/2$ and any $0<\varepsilon<1$. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking $\varepsilon=0$ is explained by the failure of solutions of the heat equation with initial data $u_0\in H^{s-1}$ to satisfy $u\in L^1(0,T;H^{s+1})$; we provide an explicit example of this phenomenon.