Well-posedness of thermal boundary layer equation in two-dimensional incompressible heat conducting flow with analytic datum

In this paper, we study the well-posedness of the thermal boundary layer equation in two-dimensional incompressible heat conducting flow. The thermal boundary layer equation describes the behavior of thermal layer and viscous layer for the two-dimensional incompressible viscous flow with heat conduction in the small viscosity and heat conductivity limit. When the initial datum are analytic, with respect to the tangential variable of the boundary, and without the monotonicity condition of the tangential velocity, by using the Littlewood-Paley theory, we obtain the local-in-time existence and uniqueness of solution to this thermal boundary layer problem.