Global regularity of three-dimensional density patches for inhomogeneous incompressible viscous flow

Toward Lions' open question in Lions (1996) concerning the propagation of regularity for density patch, we prove that the boundary regularity of the 3-D density patch persists by time evolution for inhomogeneous incompressible viscous flows, with the initial density given by (1 - eta)1(Omega 0) + 1(Omega 0c) for some small enough constant eta and some W-k + 2,W-p domain Omega(0), p is an element of]3, infinity[, and with the initial velocity satisfying some smallness condition and appropriate conormal regularities.