Lifespan of Solution to MHD Boundary Layer Equations with Analytic Perturbation of General Shear Flow

TL;DR

This paper analyzes the lifespan of solutions to the MHD boundary layer equations under analytic shear flow perturbations, revealing the magnetic field's stabilizing effect and providing a lower bound on solution lifespan.

Contribution

It establishes a lower bound on the solution lifespan for MHD boundary layer equations with general shear flow perturbations, highlighting the magnetic field's stabilizing influence.

Findings

Lifespan of solutions is at least on the order of ^{-2+} for perturbations of size .

No restriction on shear flow strength in lifespan estimates.

Magnetic field stabilizes the boundary layer dynamics.

Abstract

In this paper, we consider the lifespan of solution to the MHD boundary layer system as an analytic perturbation of general shear flow. By using the cancellation mechanism in the system observed in \cite{LXY1}, the lifespan of solution is shown to have a lower bound in the order of $\varepsilon^{-2+}$ if the strength of the perturbation is of the order of $\varepsilon$. Since there is no restriction on the strength of the shear flow and the lifespan estimate is larger than the one obtained for the classical Prandtl system in this setting, it reveals the stabilizing effect of the magnetic field on the electrically conducting fluid near the boundary.