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Toward the open question proposed by P.-L. Lions in \cite{Lions96} concerningthe propagation of regularities of density patch for viscous inhomogeneousflow, we first establish the global in time well-posedness of two-dimensionalinhomogeneous incompressible Navier-Stokes system with initial density being ofthe form: $\eta_1{\bf 1}_{\Om_0}+\eta_2{\bf 1}_{\Om_0^c},$ for any pair ofpositive 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 somesense, 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 arecompletely different from those in \cite{LZ}.

Global regularities of two-dimensional density patch for inhomogeneous incompressible viscous flow with general density