Higher regularity and asymptotic behavior of 2D magnetic Prandtl model in the Prandtl-Hartmann regime

TL;DR

This paper proves higher regularity and analyzes the long-term behavior of solutions to the 2D magnetic Prandtl model in the Prandtl-Hartmann regime, including global existence, asymptotics, and convergence to equilibrium.

Contribution

It introduces a new approach to establish higher regularity and global well-posedness for the degenerate boundary layer model, overcoming previous difficulties.

Findings

Established higher regularity of solutions.

Proved global existence and asymptotic convergence.

Demonstrated exponential decay to Hartmann layer.

Abstract

In this paper, we investigate the higher regularity and asymptotic behavior for the 2-D magnetic Prandtl model in the Prandtl-Hartmann regime. Due to the degeneracy of horizontal velocity near boundary, the higher regularity of solution is a tricky problem. By constructing suitable approximated system and establishing closed energy estimate for a good quantity(called "quotient" in \cite{Guo-Iyer-2021}), our first result is to solve this higher regularity problem. Furthermore, we show the global well-posedness and global-in-$x$ asymptotic behavior when the initial data are small perturbation of the classical Hartmann layer in Sobolev space. By using the energy method to establish closed estimate for the quotient, we overcome the difficulty arising from the degeneracy of horizontal velocity near boundary. Due to the damping effect, we also point out that this global solution will converge to the equilibrium state(called Hartmann layer) with exponent decay rate.