Global Well-posedness of a Prandtl Model from MHD in Gevrey Function Spaces

TL;DR

This paper proves the global well-posedness of a Prandtl model derived from magnetohydrodynamics in Gevrey spaces, revealing structural properties and cancellation mechanisms that ensure solution existence over time.

Contribution

It establishes the global well-posedness of a Prandtl-MHD model in Gevrey spaces with optimal index, utilizing novel cancellation techniques and structural insights.

Findings

Global well-posedness in Gevrey space with index 2

Identification of cancellation mechanisms in the Prandtl-MHD model

Structural analysis of derivative loss in the Prandtl operator

Abstract

We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer. A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index $2$. The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator