Mild Ill-Posedness in L∞ for 2D Resistive MHD Equations Near a Background Magnetic Field

Abstract

Abstract The global well-posedness on the 2D resistive MHD equations without kinematic dissipation remains an outstanding open problem. This is a critical problem. Any $L^p$-norm of the vorticity $\omega $ with $1\le p<\infty $ has been shown to be bounded globally (in time), but whether the $L^\infty $-norm of $\omega $ is globally bounded remains elusive. The global boundedness of $\|\omega \|_{L^\infty }$ yields the resolution of the aforementioned open problem. This paper examines the $L^\infty $-norm of $\omega $ from a different perspective. We construct a sequence of initial data near a special steady state to show that the $L^\infty $-norm of $\omega $ is actually mildly ill-posed.