Three-dimensional nonstationary flow of a viscous incompressible liquid is investigated in
a layer, driven by a nonuniform distribution of temperature on its free boundaries. If
the temperature given on the layer boundaries is quadratically dependent on horizontal
coordinates, external mass forces are absent, and the motion starts from rest then the free
boundary problem for the Navier–Stokes equations has an ‘exact’ solution in terms of two
independent variables. Here the free boundaries of the layer remain parallel planes and the
distance between them must be also determined. In present paper, we formulate conditions
for both the unique solvability of the reduced problem globally in time and the collapse of
the solution in finite time. We further study qualitative properties of the solution such as its
behaviour for large time (in the case of global solvability of the problem), and the asymptotics
of the solution near the collapse moment in the opposite case.