Variational inequalities of Navier-Stokes type with time dependent constraints

We consider a class of parabolic variational inequalities with time dependent obstacle of the form |u(x,t)|≤p(x,t), where u is the velocity field of a fluid governed by the Navier-Stokes variational inequality. The obstacle function p=p(x,t), imposed on u, consists of three parts, which are respectively: the degenerate part p(x,t)=0, the finitely positive part 0