In this paper, we consider in three dimensions the motion of a general inviscid, incompressible fluid with a free interface that separates the fluid region from the vacuum. We assume that the fluid region is below the vacuum and that there is no surface tension on the free surface. Then we prove the local well‐posedness of the free boundary problem in Sobolev space provided that there is no self‐intersection point on the initial surface and under the stability assumption that ${{\partial p}\over {\partial {\bf n}}}(\xi)|_{t=0}\leq -2c_{0}<0$ with ξ being restricted to the initial surface.

In this paper, we prove the global existence and uniqueness of solution to d-dimensional (for d = 2, 3) incompressible inhomogeneous Navier-Stokes equations with initial density being bounded from above and below by some positive constants, and with initial velocity u₀ ∈ H^s( R ²) for s > 0 in 2-D, or u₀ ∈ H¹( R ³) satisfying ‖u₀‖_L² ‖∇u₀‖_L² being sufficiently small in 3-D. This in particular improves the most recent well-posedness result in [ 10 Danchin , R. , Mucha , P.B. ( 2013 ). Incompressible flows with piecewise constant density . Arch. Rat. Mech. Anal. 207 : 991 – 1023 . [Crossref], [Web of Science ®] [Google Scholar] ], which requires the initial velocity u₀ ∈ H²( R ^d) for the local well-posedness result, and a smallness condition on the fluctuation of the initial density for the global well-posedness result.