Nonhomogeneous Viscous Incompressible Fluids: Existence of Velocity, Density, and Pressure

The flow of a nonhomogeneous viscous incompressible fluid that is known at an initial time $t = 0$ is considered. Such a flow is described by partial differential equations for the velocity u, the density ρ, and the pressure p, with boundary and initial conditions. The existence of a global (in time) solution u,ρ ,p for which ρ u satisfies a weak initial condition is proved. For this solution u and ρ u are not necessarily t-continuous, and $u(0)$ and (ρ u)(0) are not defined. The initial density ρ ₀ is not required to have a positive lower bound. When u₀, f, and Ω are regular, the solution is regular up to some time T_ *. For this solution, ρ u is t-continuous up to T_ * and satisfies an initial condition that is intermediate between the weak and the strong ones. If in addition ρ ₀ is not too small, but possibly zero at some points, then u is t-continuous at $t = 0$ and satisfies the strong initial conditions (ρ u)(0) = ρ ₀ u₀, u(0) = u₀, and u,ρ ,p is a global strong solution. In space dimension 2 the solution is regular for all t if ρ ₀ is bounded from below.