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

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: $η_1{\bf 1}_{\Om_0}+η_2{\bf 1}_{\Om_0^c},$ for any pair of positive constants $(η_1,η_2),$ 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, $|η_1-η_2|,$ moreover, the techniques used in the present paper are completely different from those in \cite{LZ}.

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