  We study a generalization of $T_N$-configurations, called the
$\tau_N$-configurations, for constructing certain irregular solutions of some
nonlinear diffusion systems by the method of convex integration. We construct
some polyconvex functions that support a parametrized family of
$\tau_N$-configurations satisfying a general openness condition; this will
guarantee the existence of nowhere-$C^1$ Lipschitz weak solutions to the
initial boundary value problems of the polyconvex gradient flows. We elaborate
on such constructions and the subsequent verification of the openness condition
when the dimension is at least 4 to avoid some complicated calculations that
cannot be done by hand but would otherwise be needed for dimensions 2 and 3.
