  In this paper we deal with parabolic variational inequalities of
Navier-Stokes type with time-dependent constraints on velocity fields,
including gradient constraint case. One of the objectives of this paper is to
propose a weak variational formulation for variational inequalities of
Navier-Stokes type and to solve them by applying the compactness theorem, which
was recently developed by the authors (cf. [22]). Another objective is to
approach to a class of quasi-variational inequalities associated with
Stefan/Navier-Stokes problems in which we are taking into account the freezing
effect of materials in fluids. As is easily understood, the phase change from
liquid into solid gives a great influence to the velocity field in the fluid.
For instance, in the mushy region, the velocity of the fluid is constrained by
some obstacle caused by moving solid. We shall challenge to the mathematical
modeling of Stefan/Navier-Stokes problem as a quasi-variational inequality and
solve it as an application of parabolic variational inequalities of
Navier-Stokes type.
