Vol. 2, No. 1, 2020

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

Piotr B. Mucha and Tomasz Piasecki

Vol. 2 (2020), No. 1, 123–155
DOI: 10.2140/paa.2020.2.123
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 a 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 the application of fractional spaces is irreplaceable, although the subject is a classical system.

Keywords
compressible Navier–Stokes equations, slip boundary condition, inflow boundary condition, strong solutions
Mathematical Subject Classification 2010
Primary: 35Q30, 76N10
Milestones
Received: 25 June 2019
Accepted: 27 September 2019
Published: 9 November 2019
Authors
Piotr B. Mucha
Institute of Applied Mathematics and Mechanics
University of Warsaw
Warsaw
Poland
Tomasz Piasecki
Institute of Applied Mathematics and Mechanics
University of Warsaw
Warsaw
Poland