The Dirichlet problem for steady viscous compressible flow in three dimensions

AbstractWe consider the Navier–Stokes equations for compressible isentropic flow in the steady three-dimensional case and show the existence of a weak solution for homogeneous Dirichlet (no-slip) boundary conditions under the assumption that the adiabatic exponent satisfies γ>43. In particular we cover with our existence result the cases of a monoatomic gas (γ=53) and of air (γ=75). To our knowledge it is the first result that really deals in 3-D with the existence of a weak solution in these physically relevant cases with arbitrary large external data and these boundary conditions. As an essential tool we demonstrate and use a weighted estimate respective an estimate in a Morrey-space for the pressure and resulting from this an L∞-estimate for the inverse Laplacian of the pressure