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This paper examines an averaging technique applied to the transport equationsas an alternative to vanishing viscosity. Such techniques have been shown to bevalid shock-regularizations of the Burgers equation and the Euler equations,but has yet to be applied to the similar transport equations. However, for thissystem, the classical notion of weak solutions is not always sufficient thus amore general notion of a distribution solution containing Dirac-delta functionsmust be introduced. Moreover, the distribution solution to the Riemann problemis known to be the weak-$*$ limit of the viscous perturbed transport equationsas viscosity vanishes. In comparison to the classical method of vanishingviscosity, the Riemann problem is examined for the averaged transport equationand it is shown that the same delta-shock distribution solution is captured asfiltering vanishes. Both mathematical and physical motivation are provided forthe averaging techniques considered including an existence and uniquenessresult of solutions within the class of smooth initial conditions.

A regularization for the transport equations using spatial-averaging