Steady transport equation in Sobolev-Slobodetskii spaces

We show existence of a regular solution in Sobolev-Slobodetskii spaces to stationary transport equation with inflow boundary condition in a bounded domain $Ω\subset \mathbb{R}^2$. 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.

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