Journal:Journal of Differential Equations

Key Words:Boussinesq boundary layer system; Local well-posedness; Gevrey

Abstract:In this paper, we consider the 3D Boussinesq boundary layer system in $\mathbb{R}^+\times \mathbb{R}^2$, which is a coupling of the Prandtl type equations and a thermal layer equation due to the coupling of velocity and temperature in Boussinesq equations. We observe that there is also a cancellation mechanism in the temperature equation, which have been applied to the Prandtl equation in [W. Li; N. Masmoudi and T. Yang. \newblock{\em Comm. Pure Appl. Math.} \textbf{75} (2022), no. 8, 1755--1797]. Utilizing these cancellation mechanisms and constructing good unknowns, we overcome the loss of derivative arising in not only the velocity equations but the temperature equation, then we show that the Boussinesq boundary layer system is local well-posedness in Gevrey function spaces. Furthermore, we attain the optimal Gevrey index $2$.

Indexed by:Article

Volume:450

Translation or Not:no

Date of Publication:2025-08-17