  Toward the open question proposed by P.-L. Lions in \cite{Lions96} concerning
the propagation of regularities of density patch for viscous inhomogeneous
flow, we first establish the global in time well-posedness of two-dimensional
inhomogeneous incompressible Navier-Stokes system with initial density being of
the form: $\eta_1{\bf 1}_{\Om_0}+\eta_2{\bf 1}_{\Om_0^c},$ for any pair of
positive constants $(\eta_1,\eta_2),$ and for any bounded, simply connected
$W^{k+2,p}(\R^2)$ domain $\Om_0.$ We then prove that the time evolved domain
$\Om(t)$ also belongs to the class of $W^{k+2,p}$ for any $t>0.$ Thus in some
sense, we have solved the aforementioned Lions' question %of density patch in
\cite{Lions96} in the two-dimensional case. Compared with our previous paper
\cite{LZ}, here we remove the smallness condition on the jump,
$|\eta_1-\eta_2|,$ moreover, the techniques used in the present paper are
completely different from those in \cite{LZ}.
