  We consider a class of parabolic variational inequalities with time dependent
obstacle of the form $|{\boldsymbol u}(x,t)| \le p(x,t)$, where ${\boldsymbol
u}$ is the velocity field of a fluid governed by the Navier--Stokes variational
inequality. 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 approximate
obstacle problems with everywhere finitely positive obstacles and prove an
existence result for the original problem by discussing the convergence of the
approximate problems. The crucial step is to handle the nonlinear convection
term. In this paper we propose a new approach to it.

Analysis of PDEs