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We consider a class of parabolic variational inequalities with time dependentobstacle of the form $|{\boldsymbol u}(x,t)| \le p(x,t)$, where ${\boldsymbolu}$ is the velocity field of a fluid governed by the Navier--Stokes variationalinequality. The obstacle function $p=p(x,t)$ imposed on ${\boldsymbol u}$consists of three parts which are respectively the degenerate part $p(x,t)=0$,the finitely positive part $0< p(x,t) <\infty$ and singular part$p(x,t)=\infty$. In this paper, we shall propose a sequence of approximateobstacle problems with everywhere finitely positive obstacles and prove anexistence result for the original problem by discussing the convergence of theapproximate problems. The crucial step is to handle the nonlinear convectionterm. In this paper we propose a new approach to it.

Variational Inequalities of Navier--Stokes Type with Time Dependent Constraints