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

Wei-Xi Li, Zhan Xu, Anita Yang

TL;DR

This work proves global well-posedness for the 2D Prandtl-Shercliff model in Sobolev spaces without structural data and establishes a global analytic smoothing effect in all variables, leveraging the intrinsic nonlocal diffusion from the Shercliff layer. It also analyzes a 3D linearized variant around a shear flow, proving global well-posedness for data analytic in a single tangential direction and obtaining space-time analyticity with anisotropic diffusion radii. The approach combines energy methods in anisotropic weighted Sobolev spaces for the 2D problem with a refined analytic framework using $X_r$/$Z_r$-type norms, and a weighted-Lebesgue decay theory for the 3D linearized problem, including linearly-good unknowns to capture decay. The results highlight the stabilizing influence of nonlocal Shercliff diffusion on boundary-layer dynamics and provide sharp analyticity radii that reflect anisotropic diffusion along tangential and vertical directions.

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.

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

TL;DR

This work proves global well-posedness for the 2D Prandtl-Shercliff model in Sobolev spaces without structural data and establishes a global analytic smoothing effect in all variables, leveraging the intrinsic nonlocal diffusion from the Shercliff layer. It also analyzes a 3D linearized variant around a shear flow, proving global well-posedness for data analytic in a single tangential direction and obtaining space-time analyticity with anisotropic diffusion radii. The approach combines energy methods in anisotropic weighted Sobolev spaces for the 2D problem with a refined analytic framework using /-type norms, and a weighted-Lebesgue decay theory for the 3D linearized problem, including linearly-good unknowns to capture decay. The results highlight the stabilizing influence of nonlocal Shercliff diffusion on boundary-layer dynamics and provide sharp analyticity radii that reflect anisotropic diffusion along tangential and vertical directions.

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.

Paper Structure

This paper contains 11 sections, 15 theorems, 219 equations.

Table of Contents

  1. Introduction and main results
  2. Notations
  3. Function spaces and main results
  4. Proof of Theorem \ref{['thm:2D']}
  5. Global existence and uniqueness of system \ref{['eq:2D']}
  6. Proof of Theorem \ref{['thm:2D']}: analytic smoothing effect
  7. Proof of Theorem \ref{['thm:linear']}
  8. Preliminaries: estimates in the weighted Lebesgue space
  9. Proof of Proposition \ref{['thm:heat']}: decay estimate for the heat equation
  10. Proof of Theorem \ref{['thm:linear']}
  11. Proofs of inequalities \ref{['ineq3']}, \ref{['ineq4']}, \ref{['ineq5']} and \ref{['ineq6']}

Key Result

Theorem 1.1

Assume the initial data ${u}_0\in\mathcal{H}^1(\mathbb R_+^2)$, compatible with the boundary condition in system eq:2D. Then there exists a small constant $\varepsilon_0>0$ such that if then the two-dimensional Prandtl-Shercliff system eq:2D admits a unique global-in-time solution $u\in L^\infty([0,+\infty[;\mathcal{H}^1)$ satisfying that Moreover, the solution $u$ is space-time analytic at any

Theorems & Definitions (36)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 26 more