Long time well-posdness for the 3D Prandtl boundary layer equations without structural assumption

This paper is concerned with existence and uniqueness, and stability of the solution for the 3D Prandtl equation in a polynomial weighted Sobolev space. The main novelty of this paper is to directly prove the long time well-posdness to 3D Prandtl equation without any structural assumption by the ene...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Authors: Qin, Yuming, Liu, Junchen
Format: Article
Language:English
Subjects:
Boundary layer equations
Boundary layer stability
Differential equations
Energy methods
Polynomials
Sobolev space
Structural stability
Online Access:Get full text
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Summary:This paper is concerned with existence and uniqueness, and stability of the solution for the 3D Prandtl equation in a polynomial weighted Sobolev space. The main novelty of this paper is to directly prove the long time well-posdness to 3D Prandtl equation without any structural assumption by the energy method. Moreover, the solution's lifespan can be extended to any large \(T\), provided that the initial data with a perturbation lies in the monotonic shear profile of small size \(e^{-T}\). This result improves the work by Xu and Zhang (J. Differential Equations, 263(12)(2017), 8749-8803) on the 2D Prandtl equations, achieving an extension to the three-dimensional case.
ISSN:2331-8422