Global Well-Posedness for the 2D and 3D Prandtl-Shercliff Model

TL;DR

This paper proves global well-posedness for the 2D Prandtl-Shercliff model in Sobolev spaces and demonstrates an analytic regularization effect, while establishing well-posedness for a linearized 3D version with analytic initial data in one direction.

Contribution

It establishes the first global well-posedness results for the 2D Prandtl-Shercliff model without structural assumptions and analyzes a linearized 3D model with partial analyticity.

Findings

Global well-posedness in 2D Sobolev spaces.

Analytic regularization effect in all variables.

Well-posedness for linearized 3D model with analytic initial data.

Abstract

We investigate the Prandtl-Shercliff model in both two and three dimensions. For the two-dimensional case, we establish global-in-time well-posedness in Sobolev spaces without any structural assumptions on the initial data. Furthermore, we show that the solution exhibits an analytic regularization effect in all variables, which holds globally in time and in space up to the boundary. For the three-dimensional case, we study a linearized version of the model and prove its global-in-time well-posedness for initial data that are analytic in only one tangential direction. The proofs rely crucially on the intrinsic non-local diffusion induced by the Shercliff boundary layer.