A well-posedness theory for the Prandtl equations in three space variables

2014/05/21 by Cheng‐Jie Liu, Yaguang Wang, Liu, Cheng-Jie +3 · 1 citation
Engineering · Mathematics · #35M13 #35Q35 #76D03 #76D10 #76N20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1405.5308

openalex publication_date 2014/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The well-posedness of the three space dimensional Prandtl equations is studied under some constraint on its flow structure. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the complicated secondary flow in the three-dimensional Prandtl boundary layers. And the sufficiency of the monotonicity condition on the tangential velocity field for the existence of solutions to the Prandtl boundary layer equations is illustrated in the three dimensional setting. Moreover, it is shown that this structured flow is linearly stable for any three-dimensional perturbation.

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• Unstable mode around the 3D boundary layer flow

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