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

Abstract

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 . 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′(ℝ³) for velocity with the normal component on the boundary of Ω equal to zero.

Keywords:

• Navier–Stokes equations
• compressible fluids
• weak solution

AMS Subject Classifications::

• Primary 35Q30
• Secondary 76N10

Acknowledgements

The research was supported by the University of Sud Toulon–Var, Laboratory SNC, by the Grant Agency of the Czech Republic (grant No. 201/08/0012) and by the research plan of the Ministry of Education of the Czech Republic No. MSM 6840770010.