Long time well-posedness for the 3D Prandtl boundary layer equations with a special structure

TL;DR

This paper proves long-time well-posedness for the 3D Prandtl boundary layer equations under specific structural and monotonicity conditions, extending previous local results to any large time with small initial perturbations.

Contribution

It introduces a direct energy method proof for long-time well-posedness of 3D Prandtl equations with a special structure, extending prior local well-posedness results.

Findings

Long-time existence and uniqueness of solutions under monotonicity and structural assumptions.

Solution lifespan can be extended to any large T with small initial perturbations.

The method applies to a class of 3D Prandtl equations with specific structural conditions.

Abstract

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.