Stationary compressible Navier Stokes Equations with inflow condition in domains with piecewise analytical boundaries

TL;DR

This paper proves the existence and uniqueness of strong solutions for stationary compressible Navier-Stokes equations with inflow boundary conditions in domains with piecewise analytical boundaries, using fractional Sobolev spaces.

Contribution

It establishes the existence and uniqueness of regular solutions near laminar flows in domains with boundary singularities, employing fractional Sobolev space techniques.

Findings

Existence of strong solutions in Sobolev-Slobodetskii spaces.

Boundary shape conditions are crucial for solution regularity.

Application of fractional Sobolev spaces is essential for the analysis.

Abstract

We show the existence of strong solutions in Sobolev-Slobodetskii spaces to the stationary compressible Navier-Stokes equations with inflow boundary condition. Our result holds provided certain condition on the shape of the boundary around the points where characteristics of the continuity equation are tangent to the boundary, which holds in particular for piecewise analytical boundaries. The mentioned situation creates a singularity which limits regularity at such points. We show the existence and uniqueness of regular solutions in a vicinity of given laminar solutions under the assumption that the pressure is a linear function of the density. The proofs require the language of suitable fractional Sobolev spaces. In other words our result is an example where application of fractional spaces is irreplaceable, although the subject is a classical system.