Structural stability of three dimensional steady Prandtl equation

TL;DR

This paper investigates the structural stability of three-dimensional steady Prandtl equations, addressing a longstanding open problem by introducing new analytical methods and overcoming symmetry-breaking challenges.

Contribution

It introduces a novel approach using intrinsic vector fields and maximum principles to analyze the stability of background profiles, including Blasius solutions.

Findings

Established structural stability for certain steady Prandtl solutions

Developed new maximum principle techniques for 3D boundary layer equations

Overcame symmetry-breaking difficulties in stability analysis

Abstract

The well-posedness of the three dimensional Prandtl equation is an outstanding open problem due to the appearance of the secondary flow even though there are studies on analytic and Gevrey function spaces. This problem is raised as the third open problem in the classical book by Oleinik and Samokhin [43]. This paper aims to address this open problem in the steady case by introducing a new approach to study the structural stability of background profile that includes the famous Blasius solutions. The key observations include the introduction of some intrinsic vector fields and new versions of maximum principle. In particular, we overcome the difficulties caused by symmetry breaking through the analysis on the curvature-type quantities generated by commutators of the vector fields.