Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR³₊{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ||?{r₀(·)}u₀(·)||_L²(W) + ||?u₀(·)||_L²(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.