Non-uniform Continuity for the MHD equations with only Magnetic Diffusion

TL;DR

This paper demonstrates that the data-to-solution map for incompressible MHD equations with magnetic diffusion is non-uniformly continuous in Sobolev spaces, highlighting the influence of background magnetic fields on solution stability.

Contribution

It is the first study to establish non-uniform continuity for the resistive MHD equations in Sobolev spaces, considering arbitrary constant magnetic backgrounds.

Findings

Proves non-uniform continuity of the data-to-solution map for MHD equations.

Shows magnetic background fields can stabilize solutions without losing non-uniform continuity.

Extends analysis to all Sobolev spaces with s > 0 in 2D and 3D.

Abstract

In this paper, we prove the non-uniform continuity of the data-to-solution map for the incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion in Sobolev spaces $H^s(\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Our results are first studies on the non-uniform continuity of the data-to-solution map for the resistive MHD equations. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\mathbf{B_0} \in \mathbb{R}^d$, which reveal that the strong magnetic background fields may provide the stabilization effect but still preserve the analytical feature of non-uniform continuity of the data-to-solution map.