Fujita-Kato theorem for the 3-D inhomogeneous Navier-Stokes equations (Q272276)

The authors prove the global existence and uniqueness of the solution with discontinuous density for the 3-D inhomogeneous Navier-Stokes equations with the initial data \((\varrho_0,u_0)\) in \(L^\infty({\mathbb R}^3)\times H^s({\mathbb R}^3)\) with \(s>\frac12\), and of sufficiently small \(\dot{H}^{\frac12}\) norm. Such a system describes a fluid which is obtained by mixing two miscible fluids that are incompressible and that have different densities. They also prove some decay estimates for the velocity if initially \(u_0\in L^p\) for \(p\in[6/5,2]\).