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, and the pressure p, with boundary and initial conditions. The existence of a global (in time) solution u, , p for which u satisfies a weak initial condition is proved. For this solution u and u are not necessarily t-continuous, and u (0) and (u) (0) are not defined. The initial density ₀ is not required to have a positive lower bound. When u₀, f, and are regular, the solution is regular up to some time T_ *. For this solution, 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 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 (u) (0) = ₀ u₀, u (0) = u₀, and u, , p is a global strong solution. In space dimension 2 the solution is regular for all t if ₀ is bounded from below.