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 $\rho $, and the pressure p, with boundary and initial conditions. The existence of a global (in time) solution $u,\rho ,p$ for which $\rho u$ satisfies a weak initial condition is proved. For this solution u and $\rho u$ are not necessarily t-continuous, and $u(0)$ and $(\rho u)(0)$ are not defined. The initial density $\rho _0 $ is not required to have a positive lower bound. When $u_0 $, f, and $\Omega $ are regular, the solution is regular up to some time $T_ * $. For this solution, $\rho u$ is t-continuous up to $T_ * $ and satisfies an initial condition that is intermediate between the weak and the strong ones. If in addition $\rho _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 $(\rho u)(0) = \rho _0 u_0 $, $u(0) = u_0 $, and $u,\rho ,p$ is a global strong solution. In space dimension 2 the solution is regular for all t if $\rho _0 $ is bounded from below.