Well-posedness of a generalized time-harmonic transport equation for acoustics in flow

We study a generalized time-harmonic transport equation, which appears in the Goldstein equations and allows us to model the acoustic radiation in a flow. We investigate the well-posedness of this transport problem. The result will be established under the assumption of a Ω-filling flow, which, in 2D, is simply equivalent to a flow that does not vanish. The approach relies on the method of characteristics, which leads to the resolution of the transport equation along the streamlines, and on general results of functional analysis. The theoretical results are illustrated with numerical results obtained with a Streamline Upwind Petrov-Galerkin finite element scheme.