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Seeking to develop efficient and accurate absorbing layers for the low-speed weakly compressible Navier-Stokes equations, we exploit the connection to the BGK-approximation to develop an absorbing layer with qualities similar to that of the perfectly matched layer (PML) known from linear wave problems. Representing the solutions to the BGK approximation as variations from the Maxwellian distribution allows for a direct connection to the compressible Navier-Stokes equation while leading to a linear system of equations with nonlinearities expressed in low order terms only. This allows for a direct construction of an absorbing layer with PML-like properties. We demonstrate the accuracy of the BGK approximation and the efficiency of the absorbing layer for the BGK model before pursuing this as a means of truncating the full Navier-Stokes equations with a flux based coupling between the different regions. Examples including sound propagation and vortex shedding confirms the accuracy of the approach. Although we focus on two-dimensional isentropic examples, the development generalizes to the three-dimensional non-isentropic case.