Higher order commutator estimates and local existence for the non-resistive MHD equations and related models

TL;DR

This paper proves local-in-time existence and uniqueness of strong solutions for non-resistive MHD equations in 2D and 3D, using new commutator estimates to establish uniform bounds.

Contribution

It introduces a new commutator estimate that generalizes Kato & Ponce's work, enabling the proof of local existence for non-resistive MHD equations.

Findings

Established local existence and uniqueness of strong solutions in $H^{s}$ for $s > n/2$.

Developed a new estimate generalizing Kato & Ponce's commutator estimate.

Applied the estimate to prove uniform bounds for the solutions.

Abstract

This paper establishes the local-in-time existence and uniqueness of strong solutions in $H^{s}$ for $s > n/2$ to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $\mathbb{R}^{n}$, $n=2, 3$, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato & Ponce (Comm. Pure Appl. Math. 41(7), 891-907, 1988).