Long time well-posedness for the 3D Prandtl boundary layer equations with a special structure
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914001266409472 |
|---|---|
| author | Qin, Yuming Liu, Junchen |
| author_facet | Qin, Yuming Liu, Junchen |
| contents | This paper is concerned with existence, 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-posedness to 3D Prandtl equation under monotonicity condition $\partial_{z} u >0$ and a special structural assumption $v=Ku$ $\big(\partial_{z}\big(\frac{v}{u}\big) \equiv 0\big)$ by the energy method. Moreover, the solution's lifespan can be extended to any large $T$, provided that the initial data with a perturbation lie in the monotonic shear profile of small size $e^{-T}$. This result extends the local well-posedness results established by Liu-Wang-Yang \cite{Liu-Wang-Yang-1-2017} (Adv. Math. 308 (2017) 1074-1126) and Qin-Wang \cite{Qin-Wang-2024} (J. Math. Pure. Appl. 194 (2025) 103670) for the 3D Prandtl equations to long-time well-posedness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10052 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long time well-posedness for the 3D Prandtl boundary layer equations with a special structure Qin, Yuming Liu, Junchen Analysis of PDEs 35M13, 35Q35, 76D10, 76D03, 76N20 This paper is concerned with existence, 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-posedness to 3D Prandtl equation under monotonicity condition $\partial_{z} u >0$ and a special structural assumption $v=Ku$ $\big(\partial_{z}\big(\frac{v}{u}\big) \equiv 0\big)$ by the energy method. Moreover, the solution's lifespan can be extended to any large $T$, provided that the initial data with a perturbation lie in the monotonic shear profile of small size $e^{-T}$. This result extends the local well-posedness results established by Liu-Wang-Yang \cite{Liu-Wang-Yang-1-2017} (Adv. Math. 308 (2017) 1074-1126) and Qin-Wang \cite{Qin-Wang-2024} (J. Math. Pure. Appl. 194 (2025) 103670) for the 3D Prandtl equations to long-time well-posedness. |
| title | Long time well-posedness for the 3D Prandtl boundary layer equations with a special structure |
| topic | Analysis of PDEs 35M13, 35Q35, 76D10, 76D03, 76N20 |
| url | https://arxiv.org/abs/2411.10052 |