A regularization for the transport equations using spatial-averaging

TL;DR

This paper introduces a spatial-averaging regularization for transport equations, demonstrating its effectiveness in capturing delta-shock solutions and providing mathematical and physical insights into its application as an alternative to vanishing viscosity.

Contribution

It extends averaging techniques to transport equations, establishing existence, uniqueness, and the ability to capture delta-shock solutions, which were not previously applied to this system.

Findings

Averaging captures delta-shock solutions as filtering vanishes.

Existence and uniqueness of solutions are proven for smooth initial conditions.

The method provides a valid shock-regularization alternative to vanishing viscosity.

Abstract

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