Two problems with a free boundary for the Navier–Stokes equations are considered. In the first problem, the fluid occupies a horizontal strip whose lower boundary is a motionless wall and whose upper boundary is a straight-line free boundary parallel to the wall. In the second problem, the fluid motion is rotationally symmetric. Here, the flow domain is a horizontal layer bounded by a solid plane and a parallel flat free surface. In both problems, the vertical velocity and pressure are independent of the longitudinal coordinates. In the first problem, there are three modes of motion: stabilization to a quiescent state with increasing time, blowup of the solution within a finite time, and intermediate self-similar mode in which the layer thickness unlimitedly increases with time. The same situation occurs in the second problem if the solid surface bounding the layer does not move. However, its rotation can prevent the solution collapse.