  In an earlier work we have shown the global (for all initial data and all
time) well-posedness of strong solutions to the three-dimensional viscous
primitive equations of large scale oceanic and atmospheric dynamics. In this
paper we show that for certain class of initial data the corresponding smooth
solutions of the inviscid (non-viscous) primitive equations blow up in finite
time. Specifically, we consider the three-dimensional inviscid primitive
equations in a three-dimensional infinite horizontal channel, subject to
periodic boundary conditions in the horizontal directions, and with no-normal
flow boundary conditions on the solid, top and bottom, boundaries. For certain
class of initial data we reduce this system into the two-dimensional system of
primitive equations in an infinite horizontal strip with the same type of
boundary conditions; and then show that for specific sub-class of initial data
the corresponding smooth solutions of the reduced inviscid two-dimensional
system develop singularities in finite time.
