Global well-posedness of the partially damped 2D MHD equations via a direct normal mode method for the anisotropic linear operator

TL;DR

This paper proves the global well-posedness of 2D incompressible non-resistive MHD equations with damping near a magnetic background, using a novel normal mode method that exploits anisotropy and eigenvalue analysis.

Contribution

It introduces a new normal mode method tailored for anisotropic linear operators, enabling global well-posedness results in rougher initial data spaces than previous work.

Findings

Established global well-posedness for small initial data in rougher spaces.

Developed a novel eigenvalue-based normal mode analysis for anisotropic operators.

Improved regularity requirements compared to prior results.

Abstract

We prove the global well-posedness of the 2D incompressible non-resistive MHD equations with a velocity damping term near the non-zero constant background magnetic field. To this end, we newly design a normal mode method of effectively leveraging the anisotropy of the linear propagator that encodes both the partially dissipative nature of the non-resistive MHD system and the stabilizing mechanism of the underlying magnetic field. Isolating new key quantities and estimating them with themselves in an entangling way via the eigenvalue analysis based on Duhamel's formulation, we establish the global well-posedness for any initial data $(v_0,B_0)$ that is sufficiently small in a space rougher than $H^{4}\cap L^1$. This improves the recent work in SIAM J. Math. Anal. 47, 2630-2656 (2015) where the similar result was obtained provided that $(v_0,B_0)$ was small enough in a space strictly embedded in $H^{20}\cap W^{6,1}$.