Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay

2016/04/30 by Cheng‐Jie Liu, Tong Yang, Liu, Cheng-Jie +1
Engineering · Mathematics · #35M13 #35Q35 #76D03 #76D10 #76N20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1605.00102

openalex publication_date 2016/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by the paper by D. Gerard-Varet and E. Dormy [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equations in Sobolev spaces, this paper aims to extend the result in \citeGV-D to the case when the shear flow has general decay. The key observation is to construct an approximate solution that captures the initial layer to the linearized problem motivated by the precise formulation of solutions to the inviscid Prandtl equations.

Related

• Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations
• Remarks on the ill-posedness of the Prandtl equation
• Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow
• A note on the Prandtl boundary layers
• Well-posedness of The Prandtl Equation in Sobolev Spaces