A partially strong solution to the steady Navier–Stokes equations for compressible barotropic fluid with generalized impermeability boundary conditions

We prove the existence of a partially strong solution to the steady 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 curl²u · n = 0 on the boundary. We assume that the state law for the pressure has the form for γ > 3. We call the solution 'partially strong' because only the divergence-free part of velocity and the effective pressure have regularity typical for strong solutions, while the gradient part of velocity and the density have regularity typical for weak solutions.