Blowup of solutions to the thermal boundary layer problem in two-dimensional incompressible heat conducting flow

In this paper, we study the formation of singularities in a finite time for the solution of the boundary layer equations in the two-dimensional incompressible heat conducting flow. We obtain that the first order spacial derivative of the solution blows up in a finite time for the thermal boundary layer problem, for a kind of data which are analytic in the tangential variable but do not satisfy the Oleinik monotonicity condition, by using a Lyapunov functional approach. It is observed that the buoyancy coming from the temperature difference in the flow may destabilize the thermal boundary layer.