Steady transport equation in Sobolev-Slobodetskii spaces

TL;DR

This paper proves the existence of regular solutions to the stationary transport equation in Sobolev-Slobodetskii spaces, addressing boundary singularities and applicable to complex domain shapes, with implications for fluid dynamics models.

Contribution

It introduces new existence results for stationary transport equations in Sobolev-Slobodetskii spaces under general boundary conditions, including boundary tangent points.

Findings

Existence of regular solutions in Sobolev-Slobodetskii spaces.

Applicable to domains with boundary points where characteristics are tangent.

Provides insights into boundary singularities affecting Navier-Stokes equations.

Abstract

We show existence of a regular solution in Sobolev-Slobodetskii spaces to stationary transport equation with inflow boundary condition in a bounded domain $\Omega \subset \mathbb{R}^2$. Our result is subject to quite general constraint on the shape of the boundary around the points where the characteristics become tangent to the boundary which applies in particular to piecewise analytical domains. Our result gives a new insight on the issue of boundary singularity for the inflow problem for stationary transport equation, solution of which is crucial for investigation of stationary compressible Navier-Stokes equations with inflow/outflow.