Global well-posedness of a Prandtl model from MHD in Gevrey function spaces

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.

Gevrey Well-Posedness of the Hyperbolic Prandtl Equations

“…The key observation in [8,17] is about some kind of intrinsic structure that is similar to hyperbolic feature for one order loss of tangential derivatives. Recently, inspired by the stabilizing effect of the intrinsic hyperbolic type structure, Li-Xu-Yang [22] showed the global well-posedness of a Prandtl Model from MHD in the Gevrey 2 setting. The hyperbolic Prandtl system (1.1) is more complicated in terms of loss of derivatives due to the quasi-linear term η∂ t (u • ∂ x u + v∂ y u).…”

Gevrey Well-Posedness of Quasi-Linear Hyperbolic Prandtl Equations

“…The key observation in [8,17] is about some kind of intrinsic structure that is similar to hyperbolic feature for one order loss of tangential derivatives. Recently, inspired by the stabilizing effect of the intrinsic hyperbolic type structure, Li et al [22] showed the global well-posedness of a Prandtl Model from MHD in the Gevrey 2 setting.…”