Global-in-time Stability of 2D MHD boundary Layer in the Prandtl-Hartmann Regime

TL;DR

This paper proves the global-in-time stability of solutions to the 2D MHD boundary layer equations in the Prandtl-Hartmann regime, showing magnetic effects induce damping and ensure long-term regularity.

Contribution

It establishes the global existence of analytic solutions for the 2D MHD boundary layer equations with magnetic effects, a novel result in this regime.

Findings

Magnetic diffusivity and transverse magnetic field induce linear damping.

Damping leads to global in time analytic norm estimates.

Results confirm stability of the Hartmann profile over time.

Abstract

In this paper, we prove global existence of solutions with analytic regularity to the 2D MHD boundary layer equations in the mixed Prandtl and Hartmann regime derived by formal multi-scale expansion in \cite{GP}. The analysis shows that the combined effect of the magnetic diffusivity and transveral magnetic field on the boundary leads to a linear damping on the tangential velocity field near the boundary. And this damping effect yields the global in time analytic norm estimate in the tangential space variable on the perturbation of the classical steady Hartmann profile.