A weak solvability to the steady Navier-Stokes equations for compressible barotropic fluid with generalized impermeability boundary conditions

摘要

We prove the existence of a steady solution to the Navier-Stokes equations for barotropic compressible fluid in a bounded simply connected domain with the prescribed generalized impermeability conditions u·n=0, curl u·n=0 and (curl)~2 u·n=0 on the boundary, we assume that the state law for the pressure has the form P(ρ)=ρ~γ for γ>5/3. We prove several auxiliary lemmas, e.g. on solution of the Stokes problem with the generalized impermeability boundary conditions in W~(2, p)(Ω) or on the extension of the equation of continuity satisfied in the sense of distributions from D'(Ω) to D'(R~3) for velocity with the normal component on the boundary of Ω equal to zero.