Local well-posedness in Gevrey function spaces for 3D Boussinesq boundary layer system

Abstract

In this paper, we consider the 3D Boussinesq boundary layer system in $R^{+}\times R^2$, 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 has been applied to the Prandtl equations in Li et al. (2022) [14]. Utilizing these cancellation mechanisms and constructing good unknowns, we overcome the loss of derivative arising in not only the velocity equations but also the temperature equation, then we show the local well-posedness of the Boussinesq boundary layer system in Gevrey function spaces. Furthermore, we obtain the optimal Gevrey index 2.