Some conditions for blow-up in the time-dependent Bödewadt problem

The evolution of an axisymmetric viscous rotating flow bounded by a fixed plate (Bödewadt problem) is studied. It is proved that a uniform bound on the radial velocity prevents a blow-up in the azimuthal one; the inverse holds everywhere where the radial velocity is positive. However, if the radial velocity is negative in an interval of bounded length (an inflow region) and its mean is large enough at the instant t=0t=0, a blow-up follows.