The global regularity for 3D inhomogeneous incompressible fluids with vacuum

In this paper, we investigate the global regularity for 3D inhomogeneous incompressible Navier–Stokes equations with vacuum. More precisely, we establish several Prodi–Serrin type regularity criteria in the Besov space in terms of only velocity or gradient of velocity which allow the initial density to contain vacuum. By using the embedding theory of Besov space, it is shown that our results improve the Prodi–Serrin type regularity condition which is scaling invariant in the time–space Lebesgue space.