Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids

This paper concerns the Cauchy problem for nonhomogeneous incompressible Navier–Stokes equations in the two-dimensional (2D) space. By means of the weighted initial density, we obtain the existence and uniqueness of local strong solution with vacuum as a far field state. Moreover, the solution is shown to be globally defined in time, as long as the L∞L∞-norm of a weighted density keeps bounded.