Global unique solvability of inhomogeneous incompressible Navier-Stokes equations with nonnegative density

In this paper, we consider the initial-boundary value problem to the inhomogeneous incompressible Navier-Stokes equations in Omega subset of R-2. The initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. The global existence and uniqueness of solutions are proved, for any initial data (rho(0), u(0)) is an element of (L-infinity x H-0(s)) with s > 0, which constitutes a positive answer to the question raised by Danchin and Mucha (2019 Commun. Pure Appl. Math. 72 1351-85), in which the initial velocity u(0) is an element of(1)(0) (see also Li (2017 J. Differ. Egu. 263 6512-36).