Global regularity of the three-dimensional equations for nonhomogeneous incompressible fluids

This paper is concerned with the global well-posedness of strong and classical solutions for the three-dimensional nonhomogeneous incompressible Navier–Stokes equations subject to vacuum and external forces. Let ϱ0,m0ϱ0,m0 and ff be the initial density, initial momentum and potential external force, respectively. We first show that there exists a global strong solution (ϱ,u)(ϱ,u) on R3×(0,T)R3×(0,T) for any 00μ>0 is sufficiently large, or ‖ϱ0‖L∞‖ϱ0‖L∞ or ‖|m0|2/ϱ0‖L1+‖ϱ0‖L2‖f‖L2‖|m0|2/ϱ0‖L1+‖ϱ0‖L2‖f‖L2 or ‖∇(m0/ϱ0)‖L2+‖∇f‖H1‖∇(m0/ϱ0)‖L2+‖∇f‖H1 is small enough. Although the density may vanish in some open sets, it is only assumed that u0≜m0/ϱ0u0≜m0/ϱ0 is well defined and satisfies (ϱ01/2u0,∇u0)∈L2. A uniqueness result is also proved. Next, if the given data are more regular and satisfy an additional compatibility condition used in Choe and Kim (2003) for the existence of strong solution, then the strong solution is indeed a classical one.