  We study the well-posedness and dynamic transitions of the surface tension
driven convection in a three-dimensional (3D) rectangular box with
non-deformable upper surface and with free-slip boundary conditions. It is
shown that as the Marangoni number crosses the critical threshold, the system
always undergoes a dynamic transition. In particular, two different scenarios
are studied. In the first scenario, a single mode losing its stability at the
critical parameter gives rise to either a Type-I (continuous) or a Type-II
(jump) transition. The type of transitions is dictated by the sign of a
computable non-dimensional parameter, and the numerical computation of this
parameter suggests that a Type-I transition is favorable. The second scenario
deals with the case where the geometry of the domain allows two critical modes
which possibly characterize a hexagonal pattern. In this case we show that the
transition can only be either a Type-II or a Type-III (mixed) transition
depending on another computable non-dimensional parameter. We only encountered
Type-III transition in our numerical calculations. The second part of the paper
deals with the well-posedness and existence of global attractors for the
problem.
